Struggles In Physics

Scalar Gravitational Waves in a Dark Energy Dominated Universe

In the recent years, I have been curious if the notion of “gravitational waves” ought to be enlarged once the background spacetime in question is not Minkowski spacetime, but the large scale expanding universe

$g_{\mu\nu}[\eta,\vec{x}] = a[\eta]^2 \eta_{\mu\nu}.$

In particular, the massless helicity-2 character of the gravitational wave traveling in a flat background is no longer the only allowed polarization pattern. In a previous post, I discussed the existence of spin-0 gravitational polarizations in a radiation dominated universe.

Dark Energy versus Cosmological Constant

In my recent work with Li-Ying Chou (my former undergrad, now Master’s student) and Yen-Wei Liu (my former postdoc), I turned my attention to a nearly de Sitter spacetime, engendered not by a cosmological constant in Einstein’s equations, but by a slowly rolling canonical scalar field with Lagrangian density

$\mathcal{L}_\varphi \equiv \frac{1}{2} (\nabla\varphi)^2 - V[\varphi].$

On a cosmological constant driven de Sitter geometry, the linearized Einstein’s equations have been studied quite a bit over the past decade and a half or so; and the result is that gravitational radiation does only have massless spin-2 components. However, it has been clear to me for quite some time now, that when this de Sitter-like behavior is instead due to a Dark Energy scalar–or, more generally, some extra field degrees of freedom–there will be mixing between the first order perturbations of this Dark Energy with those of the scalar metric ones. In other words, there will be a discontinuous change in the number of allowed gravitational polarization modes when comparing the two scenarios, even though the background geometry is nearly indistinguishable. For the specific case of a Dark Energy scalar driven universe, I expected massless helicity zero gravitational waves.

Traceless Tidal Forces

The key object we computed was $\delta_1 C^i_{\phantom{i}0j0}$ , the electric portion of the linearized Weyl tensor, as sourced by some isolated astrophysical system. By recalling the geodesic deviation equation, we see that these components describe the traceless tidal forces acting on a pair of infinitesimally nearby free-falling test masses. Importantly, the linearized Weyl tensor in an expanding universe is invariant under infinitesimal coordinate transformations because of conformal/Weyl invariance. This, in turn, allows us to un-ambiguously identify the “scalar” and “tensor” contributions to these traceless tidal forces.

We found that these canonical scalar Dark Energy cosmological spin-0 gravitational waves, unlike their radiation dominated cousins, propagate at the speed of light; and, in particular, the null cone spin-0 contribution to the traceless tidal force reads

(1) $\delta_1 C^i_{\phantom{i}0j0}[\text{spin}-0,\text{ null cone}] \\ \approx -\frac{G_N}{2 a[\eta] r} \left( \delta_{ij} - 3 \widehat{r}_i \widehat{r}_j \right) \mathcal{H}[\eta] \mathcal{H}[\eta_r] \sqrt{ \delta w[\eta] \delta w[\eta_r] } \left( M[\eta_r] + \frac{ \ddot{Q}_{\ell\ell}[\eta_r] }{2 a^2[\eta_r] } + \dots \right),$

where $\eta_r \equiv \eta - r$ is retarded conformal time; $a[\eta] \cdot r$ is the source-observer proper spatial distance; and $M$ is the system’s total mass. We have assumed the source is non-relativistic; and here, $\ddot{Q}_{\ell\ell}$ is the (conformal time) acceleration of the spatial trace of its mass quadrupole moment. Moreover, we have parametrized the Dark Energy equation-of-state (its pressure-to-energy density) as

$w = -1 + \delta w ,$

where the deviation $\delta w$ is presumably small; and the retarded time is $\eta_r \equiv \eta - r$ .

Of course, even though the number of polarization modes are different — as one might expect — the scalar gravitational ones are suppressed relative to their tensor cousins due to the presence of $\mathcal{H}[\eta] \equiv \dot{a}/a$ in eq. (1). On the other hand, note that this spin-0 signal is directly proportional to $\sqrt{\delta w[\eta] \delta w[\eta_r]}$ ; i.e., it is directly sensitive to the Dark Energy equation of state.

Compact Binary System

Since mass is conserved on astrophysical timescales, the high frequency part of the spin-0 traceless tidal forces in eq. (1) is the $\ddot{Q}_{\ell\ell}$ term. For a compact binary system with total mass $m$ , reduced mass $\mu$ , orbital angular frequency $\omega_a$ , and eccentricity $e$ , the high frequency portion of eq. (1) becomes

(2) $\delta_1 C^i_{\phantom{i}0j0}[\text{High freq. spin}-0,\text{ null cone}] \\ \approx -\frac{G_N^{5/3} (\omega_a \cdot m)^{2/3} \mu}{2 a[\eta] r} \frac{e}{1-e^2} \left( \delta_{ij} - 3 \widehat{r}_i \widehat{r}_j \right) \mathcal{H}[\eta] \mathcal{H}[\eta_r] \sqrt{ \delta w[\eta] \delta w[\eta_r]} \cos \psi ,$

where $\psi$ is the (retarded) orientation angle of the binary on the 2D plane they lie on. On the other hand, the spin-2 contribution reads

(3) $\delta_1 C^i_{\phantom{i}0j0}[\text{spin}-2, \text{ null cone}] \\ \approx -\frac{8G_N^{5/3} (\omega_a \cdot m)^{2/3} \mu}{2 a[\eta] r} \omega_a^2 \left( \frac{1+\cos^2\theta}{2} \cos[2(\psi-\phi)] e^+_{ij} + \cos\theta \sin[2(\psi-\phi)] e^\times_{ij} \right) ,$

where the $(\theta,\phi)$ are spherical coordinates defined by treating the orbital plane of the binary system as the $(1,2)-$ plane; and the helicity-2 polarization tensors $e^{\times,+}_{ij}$ are perpendicular to the propagation direction — i.e., the unit radial vector $\widehat{r}$ — so that $e^{\times,+}_{ij} \delta^{ij} = 0 = e^{\times,+}_{ij} \widehat{r}^i$ . By comparing equations (2) and (3), one readily recognizes the isotropic character of the former spin-0 signal. Moreover, eq. (2) is directly sensitive to the eccentricity of the orbital motion engendering the gravitational radiation.

Summary

Because we deliberately did not couple ordinary matter to the Dark Energy scalar, I expect the existence of scalar gravitational waves to be rather generic — albeit highly suppressed — in Dark Energy models of accelerated cosmic expansion, provided of course the theory itself is valid down to astrophysical length scales. I believe eq. (2) is the first concrete illustration of such spin-0 gravitational radiation emitted from the sort of compact binary systems LIGO, Virgo, etc. have been hearing from to date.

References

L.Y.Chou, Y.Z.Chu and Y.W.Liu, “Scalar Gravitational Waves Can Be Generated Even Without Direct Coupling Between Dark Energy and Ordinary Matter,” [arXiv:2310.14547 [gr-qc]].
A.Ashtekar, B.Bonga and A.Kesavan, “Asymptotics with a positive cosmological constant: I. Basic framework,” Class. Quant. Grav. 32, no.2, 025004 (2015) [arXiv:1409.3816 [gr-qc]].
A.Ashtekar, B.Bonga and A.Kesavan, “Asymptotics with a positive cosmological constant. II. Linear fields on de Sitter spacetime,” Phys. Rev. D 92, no.4, 044011 (2015) [arXiv:1506.06152 [gr-qc]].
A.Ashtekar, B.Bonga and A.Kesavan, “Asymptotics with a positive cosmological constant: III. The quadrupole formula,” Phys. Rev. D 92, no.10, 104032 (2015)[arXiv:1510.05593 [gr-qc]].
H.J.de Vega, J.Ramirez and N.G.Sanchez, “Generation of gravitational waves by generic sources in de Sitter space-time,” Phys. Rev. D 60, 044007 (1999) [arXiv:astro-ph/9812465 [astro-ph]].
Y.Z.Chu, “Gravitational Wave Memory In dS $\,_{4+2n}$ and 4D Cosmology,” Class. Quant. Grav. 34, no.3, 035009 (2017) [arXiv:1603.00151 [gr-qc]].
B.Bonga and J.S.Hazboun, “Power radiated by a binary system in a de Sitter Universe,” Phys. Rev. D 96, no.6, 064018 (2017) [arXiv:1708.05621 [gr-qc]].

Where are the wavefunctionals in the QFT transition amplitudes?

Struggles

As I’ve previously confessed, there are many seemingly basic issues in physics that I struggle to understand properly. One such issue I’m going to discuss in this post is the path integral representation of the vacuum to vacuum transition in quantum field theory (QFT) and the ensuing $i \epsilon \equiv i 0^+$ prescription for the Feynman propagator.

To begin, let us recall that in quantum mechanics, the transition amplitude from some state $| A \rangle$ to some other state $| B \rangle$ , over the time interval $[t',t]$ , is given by the path integral

(1) $\langle B[t] | A[t'] \rangle \equiv \int_{\mathbb{R}^D} d^D\vec{x} \int_{\mathbb{R}^D} d^D\vec{x}' \psi_B[\vec{x}]^* K[t,t';\vec{x},\vec{x}'] \psi_A[\vec{x}'],$

where

$K[t,t';\vec{x},\vec{x}'] \equiv \int_{\vec{x}'}^{\vec{x}} \mathcal{D}\vec{q} \int_{\prod_\infty \mathbb{R}^{D}} \mathcal{D}\vec{p} \exp\left[ i \int_{t'}^t d\tau L[\vec{q},\vec{p}] \right]$

for some appropriately defined “Lagrangian” $L$ governing the dynamics of the quantum system, that we assume is a function of position $\vec{q}[\tau]$ and momentum $\vec{p}[\tau]$ .

The key observation I wish to highlight in eq. (1) is the need to integrate the path integral $K$ itself against the initial $\psi_A$ and final $\psi_B$ wavefunctions, corresponding respectively to the initial $|A\rangle$ and final $|B\rangle$ states we are interested in. However, for some mysterious reason, when we go on to do QFT, the vacuum-to-vacuum transition amplitude somehow makes no reference whatsoever to the vacuum wavefunctional itself. For instance, the computation of the two-point function (aka the Feynman Green’s function) for a Heisenberg-picture scalar field $\varphi$ is often asserted to be given by

(2) $\langle \text{vac} |T \left\{ \varphi[x] \varphi[y] \right\} | \text{vac} \rangle = \frac{ \int\mathcal{D}\varphi e^{iS[\varphi]} \varphi[x] \varphi[y] }{\int\mathcal{D}\varphi e^{iS[\varphi]} } ;$

where the integrals over field configurations in both the numerator and denominator run over $\mathbb{R}$ at each point in spacetime and — rather crucially — do not seem to contain any explicit quantum-state dependence.

Gaussian Theory in Minkowski: Infinite Spacetime

Now, in infinite flat spacetime — i.e., examining the asymptotic past $t' \to -\infty$ to future $t \to \infty$ transition amplitude — and for a non-interacting theory with mass $m>0$ , namely

(Gaussian) $S \equiv \int_{\mathbb{R}^{D+1}} d^d x \left( \frac{1}{2} (\partial \varphi)^2 - \frac{m^2}{2} \varphi^2 \right),$

the above vacuum expectation value in eq. (2) can readily be computed within the canonical formalism, to yield

(3) $\langle \text{vac} |T \left\{ \varphi[x] \varphi[y] \right\} | \text{vac} \rangle = i \int_{\mathbb{R}^{D+1}} \frac{d\omega d^D\vec{k}}{(2\pi)^{D+1}} \frac{e^{-i\omega(t-t')+i\vec{k}\cdot(\vec{x}-\vec{x}')} }{\omega^2 - \vec{k}^2 - m^2 + i \epsilon} .$

The $i \epsilon \equiv i 0^+$ here tells us the $\omega$ integral’s contour needs to dip below the $\omega = -k$ pole and skirt above the $\omega = +k$ one; i.e., if we viewed the $\omega-$ integral as running over the reals, the poles are located at $\omega = \pm |\vec{k}| (1 - i \epsilon/(2\vec{k}^2))$ . One way to justify this contour prescription, as well as the well-known Feynman propagator result in eq. (3) is to begin with the observation that the ordinary integral

(4) $\int_{\mathbb{R}} dx e^{i\alpha x^2} = \int_{\mathbb{R}} dx e^{i\alpha_\text{R} x^2} e^{-\alpha_\text{I} x^2}$

is well defined only when the imaginary part of $\alpha = \alpha_\text{R} + i \alpha_\text{I}$ is positive; so that the integral itself is damped out in the large $|x|$ region. Next, the action of eq. (Gaussian) in Fourier spacetime,

(Gaussian.2) $S = \int_{\mathbb{R}^{D+1}} \frac{d\omega d^D\vec{k}}{(2\pi)^{D+1}} \frac{1}{2} (\omega^2-\vec{k}^2) | \widetilde{\varphi}[\omega,\vec{k}] |^2,$

tells us the scalar field $\widetilde{\varphi}$ of different momenta are decoupled from one another. Hence, up to an overall normalization, the path integral itself amounts to an infinite product of integrals of the form in eq. (4).

$\int\mathcal{D}\varphi e^{ i S[\varphi] } = \mathcal{N} \prod_{\omega,\vec{k}} \int d\widetilde{\varphi}[\omega,\vec{k}] \exp[ \frac{i}{2} (\omega^2-\vec{k}^2 + i \epsilon) | \widetilde{\varphi}[\omega,\vec{k}] |^2 ] .$

where we have now introduced an $i \epsilon$ — this amounts to rendering $\text{Im}[\alpha] > 0$ in eq. (4) — for otherwise the integral would be ill defined for each and every fixed $(\omega,\vec{k})$ . Up to a factor of 2 that I’ve not been able to hunt down thus far, I was able to show using such a Fourier spacetime calculation that eq. (2) does indeed lead to eq. (3); Peskin and Schroeder (in their path integral Chapter) does a similar calculation by placing the quantum system in a box, by doing the path integral over discrete Fourier spacetime.

To sum: For non-interacting theories, it appears to be possible to justify the apparent lack of vacuum state dependence within the path integral representation of the vacuum-to-vacuum transition amplitude, albeit in a round-about manner, if one takes the infinite past to infinite future limits.

Finite Time Propagation

Why should we, though, restrict our attention only to the asymptotic past and future? Surely we may learn more physics by demanding that the transition take place over a finite interval $[t',t]$ ? For instance, in cosmological applications, physicists have become interested in whether the perturbations imprinted in the cosmic microwave sky are due to an initial quantum state other than the vacuum one. This ‘initial state’ is usually released at a finite time, not the infinite past.

To this end, let us return to eq. (Gaussian) but take this finite range into account:

(Gaussian.3) $\int_{t'}^t d\tau \int_{\mathbb{R}^D} \frac{ d^D\vec{k} }{(2\pi)^D} \frac{1}{2} \left( |\dot{\widetilde{\varphi}}[\tau,\vec{k}|^2 + E_{\vec{k}}^2 |\widetilde{\varphi}[\tau,\vec{k}|^2 \right) ;$

with the positive energy defined as

$E_{\vec{k}} \equiv \sqrt{\vec{k}^2+m^2}.$

By expressing the action in mixed frequency-real space, we see that non-interacting field theories in infinite space are really a continuous infinity collection of simple harmonic oscillators (SHOs), with oscillation angular frequency $\Omega = E_{\vec{k}}$ , described by the Lagrangian $(1/2)(\dot{q}^2 - \Omega^2 q^2)$ . From quantum mechanics, we already know that its ground state is the Gaussian

$\langle x | E_0 \equiv \Omega/2 \rangle = (\Omega/\pi)^{1/4} \exp[-(\Omega/2) x^2]$ .

This immediately informs us that our non-interacting massive field theory has a vacuum wavefunctional

$\langle \Psi | \text{vac} \rangle = \mathcal{N}' \exp\left[ -\int_{\mathbb{R}^D} \frac{d^D\vec{k}}{(2\pi)^D} \frac{E_{\vec{k}}}{2} |\widetilde{\Psi}[\vec{k}]|^2 \right].$

Now, what we actually need to compute in QFT involves insertions of operators; namely,

$\left( \int\mathcal{D}\varphi e^{iS[\varphi]} \varphi[x] \varphi[y] \varphi[z] \dots \right)/\int\mathcal{D}\varphi e^{iS[\varphi]} .$

The well-known trick to achieve this, which facilitates perturbation theory for interacting fields, is to introduce a source in the action. Here, I shall simply sketch the construction for the quantum mechanical SHO:

$\langle E_0[t] | E_0[t'] \rangle_J \equiv \sqrt{\Omega/\pi}\int_{\mathbb{R}} dx \int_{\mathbb{R}} dx' e^{-\frac{\Omega}{2}(x^2+x'^2)} K_{\text{SHO}}[t,t';x,x'] ;$

where the SHO propagator — i.e., the path integral — is

$K[t,t';x,x'] = \mathcal{N} \int_{x'}^x \mathcal{D}q \exp\left[ i \int_{t'}^t \left( \frac{1}{2} \dot{q}^2 - \frac{\Omega^2}{2} q^2 + J \cdot q \right) d\tau \right] .$

This introduction of $J$ allows the insertion of $q[t_1] q[t_2] \dots q[t_N]$ by taking functional derivatives:

$\sqrt{\Omega/\pi}\int_{\mathbb{R}} dx \int_{\mathbb{R}} dx' e^{-\frac{\Omega}{2}(x^2+x'^2)} \mathcal{N} \int_{x'}^x \mathcal{D}q \exp\left[ i \int_{t'}^t \left( \frac{1}{2} \dot{q}^2 - \frac{\Omega^2}{2} q^2 \right) d\tau \right] q[t_1] \dots q[t_N] \\ = \frac{1}{i^N} \left. \frac{\delta^N \langle E_0[t] | E_0[t'] \rangle_J}{\delta J[t_1] \dots \delta J[t_N]} \right\vert_{J=0}.$

The $\langle E_0[t] | E_0[t'] \rangle_J$ itself can be tackled by first shifting the integration variables

$q[t' \leq s \leq t] = q_c[s] + \xi[s] + \int_{t'}^t G_s[s,\tau] J[\tau] d\tau,$

where $q_c[s]$ is the classical trajectory of the SHO that begins at $q_c[t'] = x'$ and ends at $q_c[t] = x$ ; $\xi[s]$ is the quantum trajectory that now needs to be integrated over all trajectories joining $\xi[t']=0$ to $\xi[t]=0$ , namely $\int_{x'}^x \mathcal{D}q = \int_0^0 \mathcal{D}\xi$ ; whereas $G_s[t,t']$ is the Green’s function of the SHO operator obeying Dirichlet boundary conditions. (These boundary conditions on $q_c$ , $\xi$ , and $G_s$ ensure the total $q[s]$ obeys the boundary condition $q[s=t'] = x'$ and $q[s=t] = x$ , as required by the definition of $K$ itself.) After quite a bit of work, I find

$\langle E_0[t] | E_0[t'] \rangle_J = e^{-i\frac{\Omega}{2} (t-t')} \exp\left[ \frac{1}{2} \int_{t'}^t d\tau \int_{t'}^t d\tau' (iJ[\tau]) \widetilde{G}_\text{F}[\tau-\tau'] (iJ[\tau']) \right] ,$

where

$\widetilde{G}_{\text{F}}[s] \equiv \frac{1}{2\Omega} \left( \Theta[s] e^{-i\Omega\cdot s} + \Theta[-s] e^{+i\Omega \cdot s} \right) .$

The reader already familiar with QFT would recognize this to be intimately related to the Feynman Green’s function. As we have already previously identified, going from the quantum mechanical SHO to the QFT of a massive scalar amounts to replacing $\Omega \to E_{\vec{k}}$ followed by multiplying all the relevant transition amplitudes over all momenta $\vec{k}$ . The result is

$\langle \text{vac} | T \left\{ \varphi[x] \dots \varphi[x_N] \right\} | \text{vac} \rangle = \frac{1}{i^N} \frac{\delta^N}{\delta J[x_1] \dots \delta J[x_N]} \left. \exp\left[ \frac{1}{2} \int_{\mathbb{R}^D} \frac{d^D\vec{k}}{(2\pi)^D} \int_{t'}^t d\tau \int_{t'}^t d\tau' (i \widetilde{J}[\tau,\vec{k}]) \widetilde{G}_{\text{F}}[\tau-\tau'] (i \widetilde{J}[\tau',\vec{k}]) \right] \right\vert_{J=0} .$

If we re-write the above exponent in real space and time, for e.g., $x \equiv (t,\vec{x})$ and $x' \equiv (t',\vec{x}')$ ,

$\int_{\mathbb{R}^D} \frac{d^D\vec{k}}{(2\pi)^D} \int_{t'}^t d\tau \int_{t'}^t d\tau' (i \widetilde{J}[\tau,\vec{k}]) \widetilde{G}_{\text{F}}[\tau-\tau'] (i \widetilde{J}[\tau',\vec{k}]) \\ = \int_{\mathbb{R}^{D+1}} d^d x \int_{\mathbb{R}^{D+1}} d^d x' (i J[x]) G_{\text{F}}[x-x'] (i J[x']) ,$

we uncover the Feynman Green’s function

(Feynman) $G_{\text{F}}[x-x'] = \int_{\mathbb{R}^{D}} \frac{d^D\vec{k}}{(2\pi)^D} \frac{e^{i\vec{k}\cdot(\vec{x}-\vec{x}')}}{2 E_{\vec{k}}} \left( \Theta[t-t'] e^{-iE_{\vec{k}}(t-t')} + \Theta[t'-t] e^{+iE_{\vec{k}}(t-t')} \right) .$

This result would also be obtained by performing the $\omega-$ integral in eq. (3) while ensuring the $i \epsilon$ prescription is taken into account; thereby justifying the latter by explicitly taking into account the vacuum state dependence at the ends of the time interval $[t',t]$ .

Closing Remarks

I was quite happy to see Richard Woodard recently paying tribute to the amazing Steven Weinberg, where he explained how Weinberg was one of the few field theorists who did carefully take into account the vacuum state dependence in the path integral formulation.

References

R.P. Woodard, “Big Steve and the State of the Universe,” Symmetry 15, no.4, 856 (2023) doi:10.3390/sym15040856 [arXiv:2303.05111 [hep-th]].
M.E.Peskin and D.V.Schroeder, “An Introduction to quantum field theory,” Addison-Wesley, 1995, ISBN 978-0-201-50397-5

Quantum Damped Harmonic Oscillator

Density Operator

The central object in computing statistical and quantum expectation values is the density operator, which at a specific initial time $t_0$ , is given as the following sum over all relevant quantum states $\{ | \psi_\ell \rangle | \ell = 1,2,3,\dots,N \}$ :

$\rho[t_0] \equiv \sum_{\ell=1}^N p_\ell | \psi_\ell \rangle \langle \psi_\ell | .$

These $N$ linearly independent states need not be orthogonal; whereas the $p_\ell$ describes the statistical probability that the $\ell$ -th state be `measured’; in the sense that — for some arbitrary operator $O$ — the quantum statistical expectation value is given by

$\langle\langle O \rangle\rangle[t_0] = \sum_\ell p_\ell \langle \psi_\ell | O | \psi_\ell \rangle .$

These probabilities $\{ p_\ell \}$ may parametrize partial knowledge of the quantum system at hand; or may result from `coarse graining’ the system from a more fundamental one — below, we will exploit the latter perspective. In any case, the reason why we use the density operator to compute the above expectation value, is because the latter can be expressed as a trace involving the former:

$\langle\langle O \rangle\rangle[t_0] = \text{Tr}\left[ O \cdot \rho[t_0] \right]$ .

For example, the trace of the density operator itself is

$\langle\langle \mathbb{I} \rangle\rangle[t_0] = \text{Tr}\left[ \rho[t_0] \right] = \sum_\ell p_\ell = 1 = \sum_\lambda \lambda$ ;

and the trace of the square of the density operator — aka purity — is

$\text{Pu}[t_0] \equiv \text{Tr}\left[ \rho[t_0]^2 \right] = \langle\langle \rho \rangle\rangle[t_0] = \sum_\lambda \lambda^2$ ;

where I have exploited the Hermitian character of $\rho$ to phrase its purity in terms of its eigensystem, namely, $\rho[t_0] | \lambda \rangle = \lambda | \lambda \rangle$ . Thus, we have not only shown that its eigenvalues must be bounded between 0 and 1, their sum of the squares — which yields purity itself — must also be similarly bounded:

$0 \leq \left( \text{Pu}\left[ t_0 \right] = \sum_{\lambda} \lambda^2 \right) \leq 1$ .

Time Evolution

If our quantum system were self-contained, i.e., `closed’, the time evolution of each and every $| \psi_\ell \rangle$ is simply governed by

$| \psi_\ell[t \geq t_0] \rangle = K[t,t_0] | \psi_\ell \rangle ;$

where $K$ the time evolution operator itself obeys the Schrodinger equation. With $H$ denoting its total Hamiltonian,

$i \dot{K} = H K, \qquad K[t = t_0] = \mathbb{I} .$

Its solution in the position representation is often phrased as a path integral:

$\langle \vec{x} | K[t,t'] | \vec{x}' \rangle = \mathcal{N} \int_{\vec{x}'}^{\vec{x}} \mathcal{D}\vec{q} \exp\left[ i \int_{t'}^t L[\vec{q},\dot{\vec{q}}] dt'' \right] .$

This in turn tells us, the density operator evolves in time via a pair of path integrals — one for the ket and one for the bra:

$\langle \vec{x} | \rho[t] | \vec{y} \rangle = \int_{\mathbb{R}^{2D}} d^D\vec{x}' d^D\vec{y}' \langle \vec{x} | K[t,t_0] | \vec{x}' \rangle \langle \vec{x}' | \rho[t_0] | \vec{y}' \rangle \langle \vec{y}' | K[t,t_0] | \vec{y} \rangle .$

For notational convenience, we re-phrase it as

(Time.Evol) $\langle \vec{x} | \rho[t] | \vec{y} \rangle \equiv \int_{\mathbb{R}^{2D}} d^D\vec{x}' d^D\vec{y}' K\overline{K}[t,t_0;\vec{x}',\vec{y}'] \langle \vec{x}' | \rho[t_0] | \vec{y}' \rangle ,$

where we may write the double path integrals as

$K\overline{K}[t,\vec{x},\vec{y};t',\vec{x}',\vec{y}'] = |\mathcal{N}|^2 \int_{\vec{x}'}^{\vec{x}} \mathcal{D}\vec{q}_1 \int_{\vec{y}'}^{\vec{y}} \mathcal{D}\vec{q}_2 \exp\left[ i \int_{t'}^t \left( L[\vec{q}_1,\dot{\vec{q}}_1] - \overline{L[\vec{q}_2,\dot{\vec{q}}_2]} \right) dt'' \right] .$

To reiterate: for a closed system, the $\vec{q}_{1,2}$ path integrals factorize because the bra and the ket evolve independently. However, if the system arose out of coarse graining or `tracing out’ some other degrees of freedom that is otherwise irrelevant to the problem at hand, it is then entirely possible that the $\vec{q}_{1,2}$ s become coupled. Hence, eq. (Time.Evol) still represents the time evolution of an initial density operator, but the time-evolution operator itself now reads

(In.In) $K\overline{K}[t,\vec{x},\vec{y};t',\vec{x}',\vec{y}'] = |\mathcal{N}|^2 \int_{\vec{x}'}^{\vec{x}} \mathcal{D}\vec{q}_1 \int_{\vec{y}'}^{\vec{y}} \mathcal{D}\vec{q}_2 \exp\left[ i \int_{t'}^t \left( L[\vec{q}_1,\dot{\vec{q}}_1] - \overline{L[\vec{q}_2,\dot{\vec{q}}_2]} + L_\text{IF}[\vec{q}_1,\vec{q}_2] \right) dt'' \right] .$

This doubled-path integral is usually considered part of the Schwinger-Keldysh formalism, though it should really be known as the Feynman-Vernon path integral.

As an example how such $\vec{q}_{1,2}$ coupling may arise, if we had started with an additional degree of freedom $\vec{Q}$ , where $L$ is the Lagrangian involving only the $\vec{q}$ s and $L_0$ only the $\vec{Q}$ s, while $L_1$ couples the $\vec{q}$ s and $\vec{Q}$ s; the total time evolution operator would be

$|\mathcal{N}|^2 \int_{\mathbb{R}^D} d^D\vec{X} \int_{\vec{x}'}^{\vec{x}} \mathcal{D}\vec{q}_1 \int_{\vec{X}'}^{\vec{X}} \mathcal{D}\vec{Q}_1 \int_{\vec{y}'}^{\vec{y}} \mathcal{D}\vec{q}_2 \int_{\vec{Y}'}^{\vec{X}} \mathcal{D}\vec{Q}_2 \\ \times \exp\Bigg[ i \int_{t'}^t \Bigg( L[\vec{q}_1,\dot{\vec{q}}_1] + L_0[\vec{Q}_1,\dot{\vec{Q}}_1] + L_1[\vec{q}_1,\dot{\vec{q}}_1,\vec{Q}_1,\dot{\vec{Q}}_1] \\ \qquad\qquad - \overline{L[\vec{q}_2,\dot{\vec{q}}_2]} - \overline{L_0[\vec{Q}_2,\dot{\vec{Q}}_2]} - \overline{L_1[\vec{q}_2,\dot{\vec{q}}_2,\vec{Q}_2,\dot{\vec{Q}}_2]} \Bigg) dt'' \Bigg] .$

We see the influence action arises from

$\exp\left[ i \int_{t'}^t L_\text{IF} dt'' \right] \delta^{(D)}[\vec{X}'-\vec{Y}'] \\ = \int_{\mathbb{R}^D} d^D\vec{X}\int_{\vec{X}'}^{\vec{X}} \mathcal{D}\vec{Q}_1 \int_{\vec{Y}'}^{\vec{X}} \mathcal{D}\vec{Q}_2 \\ \times \exp\Bigg[ i \int_{t'}^t \Bigg( L_0[\vec{Q}_1,\dot{\vec{Q}}_1] + L_1[\vec{q}_1,\dot{\vec{q}}_1,\vec{Q}_1,\dot{\vec{Q}}_1] - \overline{L_0[\vec{Q}_2,\dot{\vec{Q}}_2]} - \overline{L_1[\vec{q}_2,\dot{\vec{q}}_2,\vec{Q}_2,\dot{\vec{Q}}_2]} \Bigg) dt'' \Bigg] .$

The presence of the $\delta^{(D)}[\vec{X}'-\vec{Y}']$ on the LHS is due to the fact that, if we further traced over the $\vec{q}$ degrees of freedom, the result of the time-evolution $K\overline{K}$ must be an identity, so that total probability is always preserved to be unity. This argument is likely not a proof, but I believe ought to be satisfied by a non-trivial fraction of closed many- or few-body quantum systems. Note, too, that this integration over the $\vec{Q}$ may be viewed as `coarse graining’, by `averaging’ the $\vec{q}-$ dynamics over its interaction with the $\vec{Q}$ s.

To preserve probability $\text{Tr}[\rho[t \geq t_0]]=1$ for any initial density operator in eq. (Time.Evol), we therefore need

(Prob.Conserv) $\int_{\mathbb{R}^D} d^D\vec{x} K\overline{K}[t,\vec{x}=\vec{x}';t',\vec{y},\vec{y}'] = \delta^{(D)}[\vec{y}-\vec{y}'] .$

We also expect the purity to remain bounded between 0 and 1:

(Pu.Bound) $0 \leq \text{Pu}[t \geq t_0] \leq 1$ .

Quantum DHO

If we start from the outset with the following Lagrangians

$L[\vec{q},\dot{\vec{q}}] = \frac{1}{2} \dot{\vec{q}}^2 - \frac{1}{2} \left( \omega^2 - i \alpha \right) \vec{q}^2$

and

$L_{\text{IF}} = - \gamma (\vec{q}_1 - \vec{q}_2) \cdot (\dot{\vec{q}}_1 + \dot{\vec{q}}_2) - i g \vec{q}_1 \cdot \vec{q}_2$

— where $\omega>0$ and, for now, $\gamma,\alpha,g \in \mathbb{R}$ — then we will discover that this describes a damped harmonic oscillator. In particular, its one-point position expectation value is

$\text{Tr}[X \cdot \rho[t]] = \langle\langle X[t \geq t_0] \rangle\rangle \\ = \langle\langle X[t_0] \rangle\rangle \left( 2 \gamma \mathcal{G}_{\text{DHO}}[t-t_0] - \partial_{t_0} \mathcal{G}_{\text{DHO}}[t-t_0] \right) + \langle\langle P[t_0] \rangle\rangle \mathcal{G}_{\text{DHO}}[t-t_0] ,$

where

$\mathcal{G}_{\text{DHO}}[\tau] = \exp[-\gamma \cdot \tau] \sin\left[ \tau \sqrt{\omega^2 - \gamma^2} \right]/\sqrt{\omega^2 - \gamma^2}$

and, hence, $\langle\langle X[t] \rangle\rangle$ both obey the DHO oscillator equation

$\left( \frac{d^2}{d \tau^2} + 2 \gamma \frac{d}{d \tau} + \omega^2 \right) \mathcal{G}_{\text{DHO}}[\tau] = 0 = \left( \frac{d^2}{dt^2} + 2 \gamma \frac{d}{dt} + \omega^2 \right) \langle\langle X[t] \rangle\rangle$

with initial conditions $\mathcal{G}_{\text{DHO}}[\tau = 0] = 0$ , $\partial_\tau \mathcal{G}_{\text{DHO}}[\tau = 0] = 1$ , $\langle\langle X[t_0] \rangle\rangle$ and $(d/d t) \langle\langle X[t=t_0] \rangle\rangle = \langle\langle P[t_0] \rangle\rangle$ . Furthermore, we may identify the DHO retarded Green’s function as

$G^+_{\text{DHO}}[\tau] = \Theta[\tau] \mathcal{G}_{\text{DHO}}[\tau] .$

The appearance of $\gamma$ in $\mathcal{G}_{\text{DHO}}[\tau]$ , which $\langle\langle X[t] \rangle\rangle$ is built out of, indicates we must impose the non-negativity of $\gamma$ to prevent a runaway solution:

$\gamma \geq 0$ .

Next, if one proceeds to impose probability conservation in eq. (Prob.Conserv), it turns out

$\alpha = g$ ;

for otherwise the ensuing integrals may not yield the Dirac $\delta-$ functions on the RHS.

Moreover, to guarantee that purity be bounded — i.e., eq. (Pu.Bound) holds — even in the asymptotic future ( $t \to \infty$ ), one would find that

$\alpha \geq 2 \gamma \omega .$

In other words, as long as we have a damped harmonic oscillator — namely, $\gamma,\omega > 0$ — its quantization appears to require a non-zero imaginary part of its frequency-squared; and this imaginary part cannot be arbitrarily small, since it must be greater than or equal to $2\gamma \omega$ . In fact, had we set $\alpha=0$ from the start, the purity would blow up exponentially as $e^{2\gamma t}$ in the asymptotic future, violating the requirement that it stays below unity.

Finally, the variance of the position operator $\langle\langle X[t]^2 \rangle\rangle = \text{Tr}[X^2 \cdot \rho[t]]$ , in the $t \to \infty$ limit, can be shown to be completely independent of the initial density operator $\rho[t_0]$ . The infinite time limit of the density operator $\rho[\infty]$ is in fact a thermal one, provided one identifies the inverse temperature as

$\beta \equiv T^{-1} = \frac{2}{\omega} \text{coth}^{-1}\left[ \frac{\alpha}{2 \gamma \omega} \right].$

References

N. Agarwal and Y.Z. Chu, “Initial value formulation of a quantum damped harmonic oscillator,” [arXiv:2303.04829 [hep-th]].

A Message of Hope

Just 3 months ago — at the beginning of August 2022 — I got promoted from Associate Professor to Professor here at NCU: I finally have tenure! This is after 4 years of undergraduate studies; 6 years of graduate school combined with one intermediate year of hiatus; followed by 7 years of postdoctoral appointments; and, finally, 5 years as Associate Professor. A total of 23 years! I do hope this only signals a new beginning in my research and teaching career.

Fired from Graduate School

My journey grew rough soon after I got to Yale for graduate school: there is a reason why this blog is dubbed Struggles in Physics — the pain is real! Those familiar with the US system know, physics graduate school is usually a direct-PhD program, even if one is awarded a M.S. along the way. For my case, I had to file to obtain a M.S. degree because I was getting fired by my then-boss; and, hence, was being effectively kicked out of Yale if I had wanted to continue my pursuit of theoretical physics. Was I a poor performer? I have never been a terribly fast worker; but by the end of the summer I was fired, I was close to putting out my second paper on the arXiv. Moreover, I had played significant roles in both papers; independently carrying out the core calculations in the first and devising the lepton number conserving (phenomenological) neutrino density matrix equations central to the second. Academics-wise, I also remember being one of the few who passed the PhD qualifying exams during the trial run, namely, just upon reaching Yale — I should thank my education at Cal for that.

My boss at Yale was not even the person I had initially wanted to work with. I wanted to become a theoretical cosmologist, having learned cosmology itself was a science entering its growth period. There was in fact a brand-new cosmologist at the high energy theory group then; but it was only during my second year at Yale, despite having spoken to him on several occasions, that I realized he was taking for his very first PhD student someone from a different institution. Why ~~have no social life~~ work one’s butt off during 4 years of undergraduate school, so that one can get into a supposedly elite school, just to get displaced by a student from a completely different institution? And, is this sort of behavior considered professionally ethical? I will always remember the then Dean of Graduate Students at Yale Physics sniggering when this topic was briefly alluded to — I believe he knew what the cosmologist had done — when I had to speak to him about withdrawing from the program. I also wrote a letter to the Dean of the Graduate School and several folks at Yale Physics regarding the ill treatment I received from my boss; but perhaps unsurprisingly, the responses were extremely vague — American institutes are afraid of lawsuits.

Personal Responsibility

One of the key life lessons I learned during my first few years as a graduate student, was that of personal responsibility; for instance, the need to take the initiative when seeking out opportunities in a competitive field like theoretical physics. I was fortunate that Lawrence Krauss was in town to give a talk sometime during the Fall of 2006, just months after I had officially withdrawn from Yale. He agreed to meet me briefly, and when I asked if the theory group at CWRU Physics had openings, he suggested that I visit Cleveland, OH to give a talk. My short visit to CWRU gave me the impression the faculty were accessible and quite willing to work with students; and in April 2007 when Tanmay Vachaspati agreed to be my PhD advisor I turned down offers from Penn State, UCLA (whose Chair at the time told me CWRU was “not even on the map”), and University of Maryland. (NYU’s Physics Department offered me a position, but its Graduate School rescinded it, likely because it was offered too late.) I would say going to CWRU was an overall decent decision, though recent events likely mean I will not be visiting my PhD alma mater anytime soon — like many US academic institutions, it has swung to the extreme Left by adopting the anti-American DIE Religion.

One key insight into physics research I did learn from my time at Yale was the upcoming revolution in experimental gravitational wave physics and the creation of an entirely new field of gravitational-signal driven astrophysics. (This has come to pass.) Upon learning that my boss at Yale was employing quantum field theory techniques to systematize higher order calculations in the post-Newtonian program (General Relativity’s modifications of Newtonian gravity) necessary for the modeling of gravitational waveforms from inspiraling compact binary systems, I suggested developing software to, at least partially, automate the computation of the Feynman diagrams. Not surprisingly — our relationship was already rather poor by then — it was summarily dismissed. During my time at CWRU I took the deliberate effort to see to fruition my computer-automation suggestion, or at least a rudimentary version of it. I still remembered wrestling with a small section of code for several weeks, during the final stages of the project, just to make sure the 2-body portion of my second post-Newtonian calculations recovered the results in the literature. But I knew the completion of the project itself was important, so as to concretely demonstrate — i.e., personal responsibility! — that I did not leave Yale because of incompetence. It was also due to this project that gave me the expertise and experience to help out an ongoing project at UPenn during my postdoc days at ASU; this, I believe, directly led to my hiring as postdoc at Ben Franklin’s institute afterwards.

Unfortunately, my troubles did not end after leaving Yale. One of my postdocs left me rather disillusioned about the severe lack of support, opportunity for real scientific growth, and the political nature of academic dynamics. For example, I learned how it felt to be an Academic Whore when I came up with the ideas, implemented them in a long paper that took months to complete, put my boss’ name on it; only to be thrown under the bus by the same boss because he was too cowardly and too scientifically inept (relative to our competitors) to stand up for me when we were scooped. (I needed my boss’ recommendation letter; otherwise, I would have written it up as a single-author work.) Though this experience did spur me to begin pursuing my own interests — i.e., personal responsibility! — in the causal structure of waves in curved spacetimes, a topic I am still currently working on. Going it alone may not be easy; but it does reap the pleasure that comes with the freedom of inquiry.

Almost Fired Again

During my third and final postdoc at the University of Minnesota Duluth, the Dean of my College tried to fire me (likely for personal-political reasons) before my contract was up; the UMD Physics Department Chair was clearly onboard with my dismissal too. This occurred despite my teaching of mathematical methods to undergraduate physics students in the previous semester, where some of them who double-majored in math were then allowed to take a more advanced version of Linear Algebra from the math department as a result of taking my course. One would have thought that raising the bar, improving the standards of education, would earn for oneself some security against professional dismissal at an institute of higher learning. But this instead turned out to be the first time in my life I officially threatened to sue my own employer. As of this writing, I am still waiting to — though do not think I will ever — hear back from the University of Minnesota: is it OK or not OK to play politics with the education of students, and mislead them regarding the scientific validity of course content?

Hope

On the other hand, amongst those I have known over the past 23 years, many more well educated, better trained, and perhaps smarter folks than I, have since left academic physics. But here I am — tenured Physics Professor! (Being stubborn helps.) If you have a good sense that you do have the relevant chops as a physicist, but are wrestling with the realities of a tough academic job market, irresponsible supervisor(s) and/or collaborators, etc. — may I say:

Persist! Don’t give up!

Continuous Symmetries and Integral Transforms

Whenever continuous symmetry transformations — e.g., rotations, translations, scaling, etc. — are implemented on an appropriate Hilbert space, it becomes a unitary operator, expressible as

$D[\vec{\xi}] \equiv \exp[-i \xi_A J^A],$

for some set of Hermitian generators $\{ J^A | A=1,2,3,\dots\}$ and real parameters $\{ \xi^A \}$ . The eigenstates $\{ | \nu \rangle \}$ of these Hermitian operators can be used to expand arbitrary states in the Hilbert space. Namely, if $\{ | x \rangle \}$ were some original set of complete basis (usually position), completeness relations give schematically:

$\langle x | \psi \rangle \sim \sum_\nu \langle x | \nu \rangle \langle \nu | \psi \rangle$

$\langle \nu | \psi \rangle \sim \sum_x \langle \nu | x \rangle \langle x | \psi \rangle .$

As we shall see, these form integral transform pairs.

Fourier Transform and Spatial Translation Symmetry

If $|\vec{x}\rangle$ denotes a position eigenket in Euclidean space endowed with Cartesian coordinates $\{ \vec{x} \}$ , translation is defined by

$\mathcal{T}[\vec{a}] | \vec{x} \rangle = | \vec{x} + \vec{a} \rangle.$

Local translation symmetry is reflected by the position-independence of the measure in

$\langle \vec{x} | \vec{x}' \rangle = \delta^{(D)}[\vec{x}-\vec{x}'] = \langle \vec{x} + \vec{a} | \vec{x}' + \vec{a} \rangle \\ = \langle \vec{x} | \mathcal{T}[\vec{a}]^\dagger \mathcal{T}[\vec{a}] | \vec{x}' \rangle$ ,

for any constant displacement $\vec{a}$ . Global translation symmetry is reflected in the position-independence of the measure in

$\mathcal{T}[\vec{a}] \equiv \int_{\mathbb{R}^D} d^D\vec{x}' \cdot | \vec{x}' + \vec{a} \rangle \langle \vec{x}'| ;$

which is unitary for the same reason. Now, such a unitary translation operator must then be generated by a Hermitian operator $P_i$ , which is usually dubbed “momentum”:

$\mathcal{T}[\vec{a}] =\exp[ -i\vec{a}\cdot\vec{P}] .$

By identifying translation with Taylor expansion, the momentum operator is can be seen to be a derivative operator:

$\langle \vec{x} | P_i | \psi \rangle = -i \partial_{x^i} \langle \vec{x} | \psi \rangle, \qquad \forall |\psi\rangle .$

The momentum operator’s complete set of eigenstates $| \vec{k} \rangle$ are nothing but the plane waves, which read in position space as

$\langle \vec{x} | \vec{k} \rangle = \exp[i\vec{k}\cdot\vec{x}],$

where the associated eigensystem equation is

$-i \partial_{x^i} \langle \vec{x} | \vec{k} \rangle = \langle \vec{x} | P_i | \vec{k} \rangle = k_i \langle \vec{x} | \vec{k} \rangle .$

Any state $\langle \vec{x} | \psi \rangle$ , originally written in the position basis, may be expanded in momentum states — i.e., as a sum over the eigenstates of $P_i$ :

(Fourier.I) $\psi[\vec{x}] \equiv \langle \vec{x} | \psi \rangle = \int_{\mathbb{R}^D} \frac{d^D \vec{k}}{(2\pi)^D} \langle \vec{x} | \vec{k} \rangle \langle \vec{k} | \psi \rangle \\ = \int_{\mathbb{R}^D} \frac{d^D \vec{k}}{(2\pi)^D} e^{i\vec{k}\cdot\vec{x}} \widetilde{\psi}[\vec{k}] .$

The inverse transformation is

(Fourier.II) $\widetilde{\psi}[\vec{k}] \equiv \langle \vec{k} | \psi \rangle = \int_{\mathbb{R}^D} d^D \vec{x} \langle \vec{k} | \vec{x} \rangle \langle \vec{x} | \psi \rangle \\ = \int_{\mathbb{R}^D} d^D \vec{x} e^{-i\vec{k}\cdot\vec{x}} \psi[\vec{x}] .$

Equations (Fourier.I) and (Fourier.II) are of course the well known Fourier transform pairs.

Scaling Symmetry and the Mellin Transform

Scaling transformation applied to a ‘position eigenket’ $| r > 0 \rangle$ on the positive real line is defined as

$D_s[\lambda] | r \rangle \equiv \lambda^{s+1} | \lambda \cdot r \rangle .$

Here, the application-specific $s$ and the scaling $\lambda$ are both positive. If the inner product is defined as

$\langle \psi_1 | \psi_2 \rangle \equiv \int_0^\infty \langle \psi_1 | r' \rangle \langle r' | \psi_2 \rangle r'^{2s+1} d r'$

— leading immediately to the completeness relation

(Mellin.Completeness) $\mathbb{I} = \int_0^\infty | r' \rangle \langle r' | r'^{2s+1} d r'$

— the dilatation operator $D_s[\lambda]$ itself is in fact unitary. This in turn tells us, since $D_s[\lambda]$ is continuously connected to the identity, it must involve the exponential of a Hermitian generator. In fact, if we parametrize

$D_s[\lambda = e^\epsilon] | r \rangle = e^{(s+1)\epsilon} | e^\epsilon \cdot r \rangle ,$

we see the exponents add upon group multiplication:

$D_s[\lambda = e^\epsilon] D_s[\lambda' = e^{\epsilon'}] = D_s[\lambda'' = e^{\epsilon''}];$

i.e., where $\epsilon'' \equiv \epsilon+\epsilon'$ . We therefore write the operator itself as

$D_s[\epsilon] \equiv \exp\left[ -i \epsilon \cdot \mathcal{E}_s \right] .$

A direct Taylor series expansion in $\epsilon$ of $\langle \psi | D_s[\epsilon] | r \rangle = e^{(s+1)\epsilon} \langle \psi | e^{\epsilon} \cdot r \rangle$ would reveal,

$\langle r | \mathcal{E}_s | \psi \rangle = i \left( r \partial_r + s + 1 \right) \langle r | \psi \rangle$ .

This in turn allows us to solve for its eigenstate $| \nu \rangle$ in the position basis:

$\langle r | \nu \rangle = e^{-i \nu - (s+1)},$

with the normalization (defined up to constant factors) given by

$\langle \nu | \nu' \rangle = 2\pi \delta[\nu - \nu'].$

Because $\epsilon$ is a real number, $\mathcal{E}_s$ must indeed be Hermitian; and, hence, the latter’s eigenvalues are guaranteed to be real as well. At this point, we may exploit its complete set of eigenfunctions to expand any state on the half line $\mathbb{R}^+$ :

(Mellin.InverseTransform.v1) $\langle r | \psi \rangle = \int_{\mathbb{R}} \frac{d \nu}{2\pi} \langle r | \nu \rangle \langle \nu | \psi \rangle \\ = \int_{\mathbb{R}} \frac{d \nu}{2\pi} r^{-i\nu - (s+1)} \langle \nu | \psi \rangle.$

If we define

$z \equiv (s+1)+i\nu ,$

eq. (Mellin.InverseTransform.v1) may be re-phrased as an integral on the complex $z-$ plane:

(Mellin.InverseTransform.v2) $\langle r | \psi \rangle = \int_{-i \infty}^{+i\infty} \frac{d z}{2\pi i \cdot r^{z}} \langle z | \psi \rangle.$

The inverse transformation, using the completeness relation in eq. (Mellin.Completeness), is

(Mellin.Transform) $\langle \nu | \psi \rangle = \int_{\mathbb{R}^+} d r' \cdot r'^{2s+1} \langle \nu | r' \rangle \langle r' | \psi \rangle \\ = \int_{\mathbb{R}^+} d r' \cdot r'^{z - 1} \langle r' | \psi \rangle .$

The equations (Mellin.Transform.v2) and (Mellin.Transform) are of course the Mellin transform pairs. Be aware that these integrals are defined with a specific range of $s+1 \equiv \text{Re}[z]$ in mind.

References

J. J. Sakurai, Jim Napolitano, “Modern Quantum Mechanics,” Cambridge University Press; 3rd edition
Bertrand, J., Bertrand, P., Ovarlez, J. “The Mellin Transform,” The Transforms and Applications Handbook: Second Edition. Ed. Alexander D. Poularikas; Boca Raton: CRC Press LLC, 2000

A Cosmological PDE: Global Issues

In the previous post we addressed the solution of the following PDE

$\left( \partial^2 - \frac{\kappa(\kappa+1)}{\eta^2} \right) \psi[\eta,\vec{x}] = J[\eta,\vec{x}]$

by obtaining explicitly its associated Green’s function $G[x,x']$ . In this post, we shall turn to discussing global issues. First, we will generalize the PDE to

$D_x \psi[x] \equiv \left( \partial^2 - \frac{\ddot{a}[\eta]}{a[\eta]} \right) \psi[\eta,\vec{x}] = J[\eta,\vec{x}];$

where the scale factor $a[\eta]$ describes the relative size of the Universe; with the Big Bang occurring at $a[\eta=0]=0$ ; and the Universe subsequently expanding for $\eta>0$ . Keeping in mind

$D_{x} G[x,x'] = \delta[\eta-\eta'] \delta^{(3)}[\vec{x}-\vec{x}'] = D_{x'} G[x,x']$ ;

let us therefore consider the following:

$\psi[\eta>0,\vec{x}] = \int_0^\infty d\eta' \int_{\mathbb{R}^3} d^3 \vec{x}' \left( D_{x'} G[x,x'] \cdot \psi[x'] - G[x,x'] \cdot D_{x'} \psi[x'] \right) \\ \qquad \qquad + \int_0^\infty d\eta' \int_{\mathbb{R}^3} d^3 \vec{x}' G[x,x'] \cdot J[x'] .$

Integration-by-parts and the assumption that $\psi$ at spatial infinity is zero yield

$\psi[\eta>0,\vec{x}] = \int_{\mathbb{R}^3} d^3 \vec{x}' \left[ \partial_{\eta'} G[x,x'] \cdot \psi[x'] - G[x,x'] \cdot \partial_{\eta'} \psi[x'] \right]_{\eta'=0^+}^{\eta'=+\infty} \\ \qquad \qquad + \int_0^\infty d\eta' \int_{\mathbb{R}^3} d^3 \vec{x}' G[x,x'] \cdot J[x'] .$

If we further employ the retarded Green’s function, so that $G[\eta < \eta' \to +\infty] = 0$ ,

(1) $\psi[\eta>0,\vec{x}] = \int_{\mathbb{R}^3} d^3 \vec{x}' \left( G[x,x'] \cdot \partial_{\eta'} \psi[x'] - \partial_{\eta'} G[x,x'] \cdot \psi[x'] \right)_{\eta'=0^+} \\ \qquad \qquad + \int_0^\infty d\eta' \int_{\mathbb{R}^3} d^3 \vec{x}' G[x,x'] \cdot J[x'] .$

If $\eta$ had ran over $\mathbb{R}$ the first term involving a spatial volume integral at the Big Bang would not be present. Furthermore, on physical grounds, is it not strange that the field and its velocity needs to be specified at the Big Bang — where geometric tidal forces are infinite — for the field to be properly determined for $\eta > 0$ ?

Gauss’ Law Violation

What sort of trouble would arise if we had simply dropped the Big Bang initial conditions spatial integral in eq. (1)? Firstly, let us notice that, the integral of the Green’s function against the source yields the concept of the particle horizon — namely, the furthest portion of the signal from any physical and, hence, timelike source $J$ at a given time is simply the wavefront emanating from the latter at the Big Bang. Any object lying beyond the particle horizon cannot be influenced by $J$ because the integral involving it in eq. (1) is zero there.

The equations of electromagnetic and linearized gravitational fields imply a global Gauss’ law type of constraint, that the flux of the electric field (and its analogous gravitational one) across some closed 2D spatial surface at a given time, must in fact be equal to the total electric charge (or, total gravitational mass) contained within this surface. However, the particle horizon poses the following puzzle. If we had dropped the initial conditions term in eq. (1), so that the relevant electromagnetic or linearized gravitational fields due to some source $J$ is strictly zero outside its particle horizon, how is Gauss’ law obeyed if this 2D closed surface lies outside of it? In fact, this is why the initial conditions become critical for global consistency of the solutions: they must be correctly specified so that the electromagnetic and gravitational field outside the particle horizon are entirely given by the initial data in such a manner consistent with Gauss’ law.

A Cosmological PDE

In cosmologies driven by a perfect fluid with constant pressure-to-energy ration $w$ , the following wave equation appears:

(1) $\left( \partial^2 - \frac{\kappa(\kappa+1)}{\eta^2} \right) \psi[\eta,\vec{x}] = J[\eta,\vec{x}]$ .

The real parameter $\kappa$ depends on both $w$ and the dimension of spacetime $d$ ; whereas $J$ is the source of the waves $\psi$ , We are working in the conformal time coordinate system, where $\eta \geq 0$ for $0 \leq w \leq 1$ and $\eta \leq 0$ for $w=-1$ .

Solution without Fourier Transform: Nariai’s Ansatz

In such a context, it may be tempting to first transform the equation to Fourier space,

$\ddot{\psi}[\eta,\vec{k}] + \left( k^2 - \frac{\kappa(\kappa+1)}{\eta^2} \right) \psi[\eta,\vec{k}] = J[\eta,\vec{k}] .$

But, if the goal of the investigation is to understand how the signal propagates in the cosmological spacetime — what travels on the null cone, and what travels inside of it — then I would assert, in this case, that there is a better way to arrive at the answer directly in real spacetime. Specifically, I am going to describe how to obtain via Nariai’s ansatz the retarded Green’s function of the wave operator in eq. (1), obeying

(2) $\left( \partial_{\eta,\vec{x}}^2 - \frac{\kappa(\kappa+1)}{\eta^2} \right) G[x,x'] = \delta^{(d)}[x-x'] = \left( \partial_{\eta',\vec{x}'}^2 - \frac{\kappa(\kappa+1)}{\eta'^2} \right) G[x,x']$ .

Even though the $\kappa(\kappa+1)/\eta^2$ term breaks time translation symmetry, space translation symmetry is still retained. Hence, we expect that the even dimensional Green’s functions $G_{2+2n}$ can be obtained from the (1+1)D one; and the odd dimensional Green’s functions $G_{3+2n}$ can be derived from the (2+1)D one.

(3.Recursion) $G_{d+2}[\eta,\eta';R \equiv |\vec{x}-\vec{x}'|] = -\frac{1}{2\pi R} \frac{\partial}{\partial R} G_d[\eta,\eta';R].$

Nariai’s ansatz amounts to multiplying the flat spacetime massless scalar Green’s function (i.e., the $\kappa=0$ answer $\overline{G}_d[x-x']$ ) by a function that depends on spacetime solely through the object

(3.CFT) $s \equiv \bar{\sigma}/(\eta \eta') ;$

where $\bar{\sigma} \equiv (1/2)( (\eta-\eta')^2 - (\vec{x}-\vec{x}')^2 )$ is the square of the geodesic distance between $(t,\vec{x})$ and $(t',\vec{x}')$ in flat spacetime. Namely, we postulate

(4) $G_d[x,x'] = \overline{G}_d[x-x'] \cdot F_d[s].$

Keeping in mind $\partial^2 \overline{G}_d[x-x'] = \delta^{(d)}[x-x']$ and the retarded solutions

$\overline{G}_2[\bar{\sigma}] = \Theta[t-t'] \frac{\Theta[\bar{\sigma}]}{2}, \\ \overline{G}_3[\bar{\sigma}] = \Theta[t-t'] \frac{\Theta[\bar{\sigma}]}{2\pi \sqrt{2 \bar{\sigma}}};$

by inserting Nariai’s ansatz in eq. (4) into its PDE with respect to $(\eta,\vec{x})$ in eq. (2) yields in (1+1)D

(5.2D) $F_2[s] \delta^{(2)}[x-x'] + \Theta[t-t'] \left( \frac{1}{\eta\eta'} + \frac{1}{\eta^2} \right) \bar{\sigma} \cdot \delta[\bar{\sigma}] F_2'[s] \\ \qquad\qquad + \frac{\overline{G}_2[\bar{\sigma}]}{\eta^2} \left\{ s (s+2) F_2''[s] + 2 (s+1) F_2'[s] - \kappa(\kappa+1) F_2[s] \right\} = \delta^{(2)}[x-x'];$

and in (2+1)D

(5.3D) $F_3[s] \delta^{(3)}[x-x'] + \Theta[t-t'] \frac{\sqrt{\bar{\sigma}}}{\pi \sqrt{2}} \left(\frac{1}{\eta\eta'} + \frac{1}{\eta^2}\right) \delta[\bar{\sigma}] F_3'[s] \\ \qquad\qquad + \frac{\overline{G}_3[\bar{\sigma}]}{\eta^2} \left\{ s (s+2) F_3''[s] + (2s+1) F_3'[s] - \kappa(\kappa+1) F_3[s] \right\} = \delta^{(2)}[x-x'].$

$F[0]=1$ now follows from demanding the coefficients of the $\delta^{(d)}[x-x']$ on the left-hand-sides of equations (5.2D) and (5.3D) to be unity. The $\bar{\sigma} \delta[\bar{\sigma}]$ term is zero. For the remaining term proportional to $\overline{G}[\bar{\sigma}]$ to vanish, $F$ needs to obey the homogeneous equations

$s (s+2) F_2''[s] + 2 (s+1) F_2'[s] - \kappa(\kappa+1) F_2[s] = 0$

and

$s (s+2) F_3''[s] + (2s+1) F_3'[s] - \kappa(\kappa+1) F_3[s] = 0.$

Nariai’s ansatz has thus reduced a PDE into an ODE with known analytic solutions.

Results

Altogether, we find the Legendre function $F_2[s] = P_\kappa[1+s]$ and the associated Legendre function $F_3[s] = (s/(2+s))^{1/4} P_\kappa^{(1/2)}[1+s].$ That the solution is unique is because the other linearly independent solution involves $Q_\kappa[1+s]$ for even $d$ ; and $Q_{\kappa}^{(1/2)}[1+s]$ for odd $d$ . Both are singular as $s \to 0$ .

For even dimensions, eq. (3.Recursion) therefore reads

(6.Even) $G_2[x,x'] = \Theta[t-t'] \frac{\Theta[\bar{\sigma}]}{2} P_\kappa[1+s] \\ G_{2+2n}[x,x'] = \Theta[t-t'] \left( \frac{1}{2\pi} \frac{\partial}{\partial \bar{\sigma}} \right)^n \left( \frac{\Theta[\bar{\sigma}]}{2} P_\kappa[1+s] \right).$

For odd dimensions, eq. (3.Recursion) in turn reads

(6.Odd) $G_3[x,x'] = \Theta[t-t'] \frac{ \Theta[\bar{\sigma}] }{ 2\pi \sqrt{2 \bar{\sigma} } } \left( \frac{s}{s+2} \right)^{\frac{1}{4}} P_{\kappa}^{(1/2)}[1+s] \\ G_{3+2n}[x,x'] = \Theta[t-t'] \left( \frac{1}{2\pi} \frac{\partial}{\partial \bar{\sigma} } \right)^n \left( \frac{ \Theta[\bar{\sigma}] }{ 2\pi \sqrt{2 \bar{\sigma} } } \left( \frac{s}{s+2} \right)^{\frac{1}{4}} P_{\kappa}^{(1/2)}[1+s] \right).$

Four Dimensions

In (3+1)D we may witness how Nariai’s ansatz yields a clean split between light cone versus tail propagation without doing any Fourier transformation; i.e., eq. (6.Even) readily informs us:

(6.4D) $G_4[x,x'] = \frac{\Theta[t-t']}{4\pi} \left( \delta[\bar{\sigma}] + \Theta[\bar{\sigma}] \frac{P'_\kappa[1+s]}{\eta \eta'} \right),$

where we have used $P_\nu[1] = 1$ . The $\delta[\bar{\sigma}]$ describes propagation on the Minkowski light cone; in fact, we have the (hopefully familiar) retarded conditon $\Theta[t-t'] \delta[\bar{\sigma}] = \delta[t-t'-R]/R$ . Whereas, the expression involving the derivative of the Legendre function, $P'_\kappa[1+s]/(\eta\eta')$ , describes the detailed inside-the-null-cone propagation of the waves.

Speculations

Is there a deeper meaning to Nariai’s ansatz? Note that the object $s$ in eq. (3.CFT) is conformally invariant from a higher dimensional point-of-view. Namely, by viewing the $d-$ coordinates $(\eta,\vec{x})$ and $(\eta',\vec{x}')$ as residing on a de Sitter-like hyperboloid in one higher dimensional flat spacetime, one may in fact derive $s$ as a Lorentz ‘dot product’ between the two points.

Reference

Y.Z. Chu, “More On Cosmological Gravitational Waves And Their Memories,” Class. Quant. Grav. 34, no.19, 194001 (2017) arXiv:1611.00018 [gr-qc]].
H. Nariai, “On the Green’s Function in an Expanding Universe and Its Role in the Problem of Mach’s Principle,” Prog. Theor. Phys. 40, 49 (1968).

The College Fix Interview

Some time in December 2021, Daniel Nuccio from The College Fix interviewed me over email. The article just appeared a couple of days ago — see below. I appreciate it very much that Daniel used the acronym I like to use: DIE, and not DEI or any other ordering of the 3 letters.

I sincerely hope more scientists — especially the senior ones — would pick up their ovaries and balls and push back strongly against the DIE Religion. Meritocracy, Reason, and Scientific Integrity are foundational to a robust and free society.

Scientists must not ‘cower to the latest political ideologies’: Physicist challenges ‘DIE’ dogmas

Update, 31 March 2022: I wrote to 30+ members of the CWRU Board of Trustees to complain about getting censored by CWRU Physics, not because I have a pressing need to view its Twitter pages, but to hold my alma mater accountable to its own “Freedom of Expression” (Chicago-like) statement. Unfortunately I received very few replies; though I was told by the Office of President Eric Kaler that I would be receiving a response from the Office of the Provost.

On the other hand, I had an exchange with one Board member, where his final message read as follows.

Yi Zen – this is my last reply and it is from me personally – not from CWRU. With courtesy your complaint simply does not deserve this much attention. You are right, I have not read your comments carefully, and I do not intend to. …… I’m sure you are academically smart but you are demonstrating a form of social stupidity. Many years ago I graduated phi beta kappa in physics, so I value academic excellence but I have also learned the limits of academic excellence. We need diversity at CWRU, and we have no interest in a public debate on the value of diversity… at least not with someone who seems not to be open minded. CWRU has more important issues than yours and time is limited. The university is doing you a favor by not distributing your racist comments. There are many kinds of merit and intelligence. As you become older and wiser you will learn this. Your definition of merit is simply way too narrow. Bottom line – free speech is not freedom to attack others. I suggest that you focus your attention on issues that are more constructive. Free speech is not freedom to shout “fire” in a crowded theater. You are trying to start a fight on this and we have no interest in fighting. ……

I made sure to let this man know, not only does he not deserve to be on the Board of Trustees, he also just proved why I am in fact fighting for the freedom of expression to be respected at CWRU.

Cherenkov Gravitational Radiation

Yen-Wei Liu and I have been trying to understand whether the Bardeen gauge-invariant scalar metric perturbations in cosmological backgrounds ought to be understood as gravitational radiation — wave solutions capable of carrying energy-momentum to infinity, arising from the scalar sectors of Einstein’s equations linearized about a radiation dominated universe, in particular. Thus far, we have gathered the following observations pointing to the answer being in the affirmative.

The Bardeen scalars obey a wave equation with respect to a modified ‘cosmological metric’ with acoustic speed $1/\sqrt{3}$ , after their equations-of-motion are decoupled from the vector and tensor ones. This result is well-known, and can be found in Chapter 7 of Mukhanov’s cosmology text. What appears less understood is whether the corresponding solutions are to be interpreted as radiation.
To this end, in arXiv: 2001.06695, Yen-Wei and I first computed the gauge-invariant linearized Weyl tensor in a radiation-dominated universe, using the homogeneous solutions of the Bardeen scalars, and found a non-trivial result. Since the Weyl tensor encodes the ‘trace-less’ portion of the physical tidal forces induced by spacetime curvature, this indicates Bardeen scalars not only can propagate freely but are capable of squeezing and stretching — i.e., doing work upon — material objects immersed in such a cosmological geometry.
In the same arXiv: 2001.06695, Yen-Wei and I also computed the Bardeen scalars’ contribution to the linearized Weyl tensor sourced by some hypothetical astrophysical source. As expected on physical grounds, in the far zone, the form of this inhomogeneous solution approaches that of the homogeneous solutions — outgoing spherical waves become, approximately, plane waves. This, in turn, allowed us to not only further corroborate this Bardeen-scalars-can-do-work interpretation; we were able to extract the corresponding polarization and oscillatory (spin-0-like) patterns in the short-wavelength limit.

**Leftmost and middle panels**: The (usual) ‘spin-2’ tensor gravitational wave polarization patterns.
**Rightmost panel**: Scalar gravitational wave polarization pattern in a radiation dominated universe.

That the Bardeen scalars obey a wave equation with acoustic speed $1/\sqrt{3}$ suggests that — if they can be indeed regarded as gravitational radiation — Cherenkov gravitational radiation is possible in the radiation era of our universe. This prompted our more recent work in arXiv: 2108.13463.

Cherenkov gravitational radiation due to a supersonic source via Huygens principle.
Figure from arXiv: 2108.13463.

In electromagnetism, Cherenkov radiation occurs in a medium whenever an electrically charged particle is moving faster than the medium’s effective speed of light. The electromagnetic signal’s wave front cannot outrun the charged particle. Hence, a shockwave forms because the electromagnetic fields are strictly zero outside of it.

What Yen-Wei and I argued was that an analogous situation occurs whenever primordial black holes or cosmic strings — if they exist! — moves through the photon’s relativistic fluid at speeds greater than $1/\sqrt{3}$ . Specifically, we computed the Bardeen scalars’ contribution to the linearized Weyl tensor for a point mass and an infinitely straight Nambu-Goto wire moving along a supersonic geodesic, and derived the shape and strength of their Cherenkov shockwave. Near the Cherenkov cone or wedge with unit normal $\widehat{u}$ , we argued that tidal forces acting on a pair of free falling test masses oriented along $\widehat{\xi}$ would be proportional to $1-3\widehat{\xi}\cdot\widehat{u}$ . Furthermore, for the supersonic point mass, this tidal force would blow up as $1/\sqrt{\perp\text{ distance to Cherenkov shock front}}$ . Once again, that such a Cherenkov shock front can exert geometry-induced work indicates Bardeen scalars constitute gravitational radiation.

I pose a question for future work:

If primordial black holes or cosmic strings exist, and if they end up moving faster than $1/\sqrt{3}$ during the radiation era, are there signatures of this Cherenkov gravitational radiation that are still observable at our current epoch?

Update, 31 March 2022: I added an exaggerated visualization of the gravitational wave polarization patterns.

Are Some Physicists More Equal Than Others at CWRU?

On September 10, I sent the following letter — with more personal aspects removed — to Case Physics faculty to express my deepest disappointment that my alma mater has adopted DIE (Diversity, Inclusion, Equity).

To Current and Emeritus Faculty at my alma mater:

Some of you may not know who I am, so let me briefly introduce myself. I graduated with my PhD from Case in 2010, working with Tanmay Vachashpati (who is now at ASU) and Glenn Starkman. Today, however, I am not writing as Glenn’s (nor Tanmay’s) former PhD student, but rather as an alumnus deeply disturbed by the embracing of DIE (Diversity, Inclusion, Equity) at my alma mater.

I wish to state in unequivocal terms: I am vehemently opposed to DIE, because it is based neither on scientific evidence nor reason, but on the far Left ideology of equal representation (or, equal outcomes) — as opposed to equal opportunities. By recruiting more Blacks, women, etc. into our physics programs for the sake of equal representation, i.e., just because we do not see 13% of physicists being Black or 50% women, is not only misguided but will ultimately eat away our scientific, academic, and professional standards. Science is about seeking the truth, understanding reality as it is; and should therefore not be corrupted by our own ideologies. Yet, this is what Academia and, very sadly, my alma mater appears to be doing. Given there is always a finite amount of resources, artificially increasing certain groups’ representation will not only lead to perverted reward structures; but also end up punishing the hardworking and competent.

In many ways, DIE has already been in place in academia for several decades: it’s otherwise known as affirmative action. The SAT, GMAT, LSAT, and MCAT scores of admitted applicants to college, business, law and medical schools are significantly skewed. Just because an American is born Black means he/she would have a marked advantage over his/her fellow American who happens to be born White or Asian. Note that the Black academic achievement gap is real — the data is there for all to examine — and no amount of affirmative action is going help improve their existing under development, if we bury our heads in the sand and refuse to look into other factors. If “too many Blacks” sounds racist to you, so should “too many Whites”. As I understand it, during the first half of the last century, higher education institutes like Harvard, Princeton, etc., regularly said “too many Jews” to impose an artificial upper limit to curb Jewish(-American) admission; these days, it is Asian-Americans. We do not seem to have learned from history; that we should simply reward folks based on hard work, competence, and potential; regardless of their backgrounds or physical characteristics.

This email will be blunt, and you will understand why by its end.

I am growing extremely concerned that Academia is discarding values that are most conducive to the flourishing of Science — meritocracy; intellectual, academic, and scientific integrity; freedom of inquiry; freedom of expression; and freedom of conscience — in favor of Social Justice with Capital Letters. Even our (astro)physics communities have become deeply corrupted, as has been made clear and rather public by the extreme-Left Particles for Justice (PfJ) movement, signed by thousands of our virtue signaling self-righteous colleagues. I note with tremendous dismay a number of faculty from my alma mater have endorsed it too, despite its latest petition explicitly demanding that meritocracy be challenged and some mystical “White Supremacy” be urgently dismantled. (What sort of funny White Supremacist Nation is the US when, for instance, a Black Commander-in-Chief is democratically elected twice; and Whites as a group are currently out-earned by quite a number of other ethnic/racial groups?) Why did even the American Physical Society (APS), a scientific organization, turn to political activism by jumping onto the PfJ bandwagon and endorsing the far Left #Strike4BlackLives and #ShutDownSTEM movements last year? Why does CWRU Physics have “Black Lives Matter” on its department webpage; why is it getting so blatantly political — and, for that matter, does it thoroughly understand what it is endorsing? (Factually speaking, unarmed Blacks are killed by cops at the rate of 10-20 per year. This is certainly not a major issue of violence against Blacks, especially compared to the far more serious issue of violence, on the order of thousands of incidents per annum, within the Black community — see, Chicago, for instance — which disturbingly receives comparatively little attention.) Importantly: when scientific organizations such as APS or Case Physics turn political — I should really say, turn Woke — we should not be surprised that this leads to an erosion of the public’s trust in Science. And Trust, once lost, will be very difficult to earn back.

To be clear, as individuals residing in free societies, we should all have the freedom to endorse or condemn any political movements we desire. But when acting on behalf of Science, we scientists have a responsibility to stick as much as possible to reality, evidence, reason, logic, etc.; however strongly we personally feel about political or moral issues. Scientific organizations like the APS or Case Physics should stay as apolitical as possible. The first petition of PfJ claimed to be for Science, but it was really an extreme Left mob trying to cancel Alessandro Strumia, who delivered a talk at CERN regarding his scientific work on bibliometrics; specifically, citations of female versus male high energy physicists. Not only have two anonymous colleagues of ours since written up rebuttals, showing flaws in their arguments; much more crucially, Strumia’s work has since been published [StrumiaApr21] — likely the only person from that HEP+Gender CERN conference who did so. Just to highlight a scientific result: he found M vs F citations to be consistent with what is known in the literature as Higher Male Variability, that for certain traits like intelligence, males as a group tend to exhibit a higher variance than females. If true, Higher Male Variability implies, for e.g., there will be significantly more male than female Nobel Laureates and winners of Darwin Awards. This is a scientific issue: if you think the evidence for it is flawed, by all means write papers debunking it. But just the discussion of such topics got Lawrence Summers cancelled years ago; and James Damore fired by Google more recently. And now, this anti-science intolerance has, too, corrupted our (astro)physics communities.

Where is our respect for the freedom of expression and inquiry, not to mention our collective scientific integrity? Freedom of speech/expression and inquiry are not frivolous luxuries: within the scientific context, it ensures all ideas can be heard and pursued; so as to be compared for their merits and faults, however difficult or taboo the subject is. This in turn ensures we may collectively acquire a more comprehensive and accurate picture of reality itself; i.e., how or what Nature or the human condition actually is. Moreover, it should be Scientific Integrity 101, that we scientists are not allowed to censor or shout down facts just because we dislike them.

While folks like Strumia who were doing real science are vilified, I see the players of this game of far Left Identity Politics widely lauded and rewarded. Jessica Wade, the physicist who launched the first salvos against Strumia, did so on Twitter and through her article on New Scientist by comparing his talk to Damore’s “Google Memo” and by asserting it was “unevidenced”, as if that would instantly discredit Strumia’s points right away. Instead she showed herself to be not only dishonest — Strumia’s talk was evidenced; he did a whole bunch of bibliometrics and even got published — but also lacked a basic understanding of the scientific validity of Damore’s Memo. (Damore had a background in biology, and hence knew the relevant literature.) Unfortunately, not only did PfJ continue Wade’s volley against Strumia; she even appeared on “Nature’s 10”, where part of the citation praised her attack against him. (Who runs Nature these days?) Another key player is physicist Chanda Prescod-Weinstein, the beyond-extreme Left leader of PfJ, who openly called Strumia racist, even though he did not discuss race at all. I am absolutely appalled that my alma mater recently advertised one of her identity politics driven talks on the Department’s Facebook page — CWRU Physics is not the far Left wing of the Democratic Party, and should stick strictly to physics. Prescod-Weinstein even received ~$100,000 from physicists-run FQXi to work on her crackpot “White Empiricism” paper [CrackPot1], deliberately trying to obfuscate and entangle her far Left fundamentalism with physics. (My guess is, from articles I’ve read, her recent book is no better.) She also wrote a Slate article [CrackPot2] titled ‘Stop Equating “Science” With Truth’; and sub-titled “Evolutionary psychology is just the most obvious example of science’s flaws”; where she too showed her own ignorance of the validity of Damore’s Memo. (Has she not heard, evolution is well established science?) Like Wade, Prescod-Weinstein also appeared on “Nature’s 10”; which praised her for the PfJ, #Strike4BlackLives and #ShutDownSTEM activism. Furthermore, she was given the 2021 Edward A. Bouchet Award by the APS, which also praised her for PfJ. This same award was given to Juan Maldacena (of AdS/CFT) back in 2004; I consider it a grave insult to Maldacena, that the APS has stooped so low to pander to far Left ideology.

Let me emphasize this is not primarily about Wade nor Prescod-Weinstein (nor even Strumia) per se. Rather, it is clear to me, without the support of the scientific communities at large, dishonest players such as Wade and Prescod-Weinstein who are openly willing to corrupt science with identity politics would not have garnered so much positive attention and praise. Even CERN — currently the place on the planet for particle physics — appeared to have bowed to radical feminism and extremely illiberal elements within our scientific communities when it booted Strumia. Despite its publication, Strumia’s scientific paper on M vs F physicists’ bibliometrics was also prohibited by the arXiv; even though dubious identity politics related papers were allowed. I’m very saddened to say: when it comes to far Left Ideology, the collective Emperor of our scientific communities — including (astro)physics — has no clothes.

On the Left, where Academia firmly belongs — the data says it’s roughly Democrats : Republicans = 6:1 in Physics — we often enjoy ridiculing the Right for science denial, particularly regarding the climate. Some of the criticism is accurate, but I believe it is way overdue to examine the plank in our own eyes. It is of course likely there are individual cases of discrimination or harassment against our fellow physicists on the basis of sex, color, and even other categories that have thus far gone unidentified. But on the macro scale, it is an utter failure of both intellectual rigor and scientific skepticism to not recognize that disparity does not necessarily imply discrimination; all relevant factors ought to be considered.

One key factor that is commonly and oftentimes deliberately neglected by the Academic Left is: we humans are not blank slates. Like all life forms on Earth, our biological makeup, including the brain, are subject to powerful forces of evolution. For instance, just because there are roughly M:F = 1:1 in the overall human population does not imply, if there isn’t M:F = 1:1 in physics, there must be rampant discrimination. As I understand it, differences in prenatal testosterone exposure have been shown to affect M vs F interests; women tend to prefer “people” and men “things”. (See, for e.g., [PinkerYoutube]; which includes a discussion of Higher Male Variability.) In fact, the M vs F career distributions in the US quite clearly reflects this. How often do you hear folks complaining about sexism against women in truck driving (dominated by men), or against men in nursing (dominated by women)? Even though women did face significant institutional obstacles in the past to higher education, the West has made tremendous progress over the past decades. In the US and since the 1980s, women on the whole already have more access than men to higher education — and the gap is still growing. Furthermore, high performing women oftentimes have broader interests than their male counterparts; hence, such women actually have more options, not fewer. Despite these, we still see female-only scholarships and (faculty) positions, etc. in Academia, sometimes in direct violation of Title IX anti-discrimination laws. Openly pointing to these facts these days readily gets one labeled as “sexist”; nevermind whether or not they are true.

Humans spend up to 1/4 (or more!) of their lives with their parents, yet we neglect the importance of the family when it comes to life outcomes such as educational attainment. If Academia did not have such strong biases against the Right — remember thought diversity? — but open their minds and listen to Black conservatives like Thomas Sowell, Walter Williams, Glenn Loury, Larry Elder, Jason Riley, etc., we will learn that the Black family remained largely intact and their economic conditions did in fact improve considerably after the abolishment of slavery and through the Jim Crow days. But upon the implementation of the Welfare State after the Civil Rights Movement of the 1960s, many Black women were incentivized to become single mothers. The rate of Black kids born to single moms has since risen from ~25% to the current ~70%. Is it surprising that many Black youth, especially boys, get into serious trouble and become disconnected from proper education? (Even some well known Black rappers and hip hop artists are involved in gangs.) In many places, studying hard is ridiculed as “acting White”. Of course, the lack of competitive and good quality K-12 schools, especially in poor neighborhoods, does not help either. No amount of DIE effort at the (under)graduate level is of relevance to all these points I’ve raised.

At the same time, there have been brave researchers such as Richard Herrnstein, Charles Murray, James Flynn, Arthur Jensen, etc. who dared ask if there are cognitive differences between racial groups. My guess is, many in academia would likely be extremely uncomfortable with this sort of questions; or immediately cry racism. It may not be well known to physicists, but the American Psychological Association wrote a consensus report in the 1990s acknowledging there are in fact average intelligence differences between groups. For instance, the Black-White IQ gap is roughly 15 points [APAReport]. (As I understand it, the causes of these group-level differences are still being hotly debated and investigated; but as already alluded to, such research has been made extremely taboo by Academia and the science-denying Left in general.) If there are indeed racial differences in average intelligence, this would be an important factor in understanding why certain groups are currently “over” or “under” represented in cognitively demanding disciplines such as (astro)physics. In fact, the higher the intellectual threshold of a given activity/career/job, the more sensitive it would be to any such average differences: this can be seen from an asymptotic analysis of the tails of the Gaussian distribution. In any case, these are scientific questions, and should be approached objectively and meticulously like any other scientific question.

A closely related incident occurred recently. Theoretical physicist Steve Hsu got demoted after a Twitter mob was launched against him, led by a graduate student at his school; all because of Hsu’s research into the relation between genetics and IQ/intelligence. While thousands of our colleagues thump their chests in faux SJW-righteousness to cancel Strumia for doing politically incorrect science, far fewer stood up to defend Hsu’s academic freedom and the freedom of inquiry. I saw only a handful of (astro)physicists signing the petition in support of him; while even some of his own faculty colleagues at Michigan joined the mob against him. That his school’s President bowed to the wishes of an online mob only speaks to how cowardly and unprincipled senior academic administrators have become. The vast disparity in the responses against Strumia versus those for Hsu by our own (astro)physics colleagues is an indication that we are not making any pro-Science progress whatsoever, despite what PfJ may try to sell us, but rather we are regressing extremely badly. Simply put, I fear Academia — including the (astro)physics disciplines — is losing its foundational values.

Now, it should be pointed out, just because groups differ in standard deviation or average performance — whatever the underlying cause (cultural, environmental, genetic/biological, etc.) — does not imply one group should be treated better than the other. Morality should be cleanly divorced from scientific facts, but it is not my place to come between you and your personal conscience / rabbi / priest / pastor / etc. What is of relevance to the discussion at hand is, despite differences in standard deviation and means, there will be highly capable folks admitted from every group, albeit in different proportions, as long as our system remains strictly unbiased and merit-based. Moreover, Western civilization has long figured out, to treat our fellow humans fairly, we should not judge folks based on the average (or worst) characteristics of their racial groups, sex, etc.; but, quite crucially, we should instead judge one another on a case-by-case basis — namely, as individuals. However, what DIE does is in fact the reverse. Academia’s incessant obsession with “identity” will only prove to be not only anti-meritocratic but also divisive, because it is encouraging people to judge each other by irrelevant but highly immutable traits, reinforcing our tribal nature.

While we enjoy promoting “Diversity and Inclusion” these days; it is apparent to me this is only skin-deep. Instead of judging our fellow scientists by the content of their scientific output, competence, and potential; we judge them by their sex, race, and other irrelevant (and, oftentimes, immutable) “identities” or characteristics to label them as “diverse”. What is of relevance to Science is thought or intellectual diversity, but here our hypocrisy is stark: any slight deviation from the far Left narrative approved by Academia is summarily sent to Gulag 13 — as was the case of Strumia, Summers, and many more. How “inclusive”!

In early April of this year, I noticed the co-chairs of my alma mater have openly and rather publicly endorsed the formation of a DIE-based physics student group PURMS. The student group had posted Corbin Covault and Glenn Starkman’s endorsement email on Twitter. I knew it would be risky, given the current very charged political climate within Academia; but given how public the endorsement was, I expressed my deep concerns on the thread and also directed a separate message towards Glenn. Despite (repeated) use of the words “meritocracy”, “professional and scientific standards”, “scientific integrity” — and despite Glenn being my PhD co-advisor, whom I’ve known for well more than a decade — I was met with the following highly public condemnation:

“Your mentors and advisors are horrified, troubled, and concerned by your posts, many of which have only recently come to our attention.”
Glenn Starkman, Apri 4, 2021 (on Twitter)

I even tried to back up my concerns with concrete data available right there on Twitter; regarding the SAT, MCAT, and LSAT scores already alluded to above. But before I could even press “reply” Glenn had already blocked me. Soon after, I discovered I was blocked by several of Case Physics Twitter accounts — CWRU PURMS and the CWRU Physics page itself; as well as by Ben Monreal.

[Over email, I was accused of “attacking” and “delegitimizing” students.]

[A few paragraphs dropped.]

I will let readers judge for themselves if I had actually attacked anyone — see link below [Blocked], if you’re interested in verifying the facts for yourself. Here, I will simply and directly falsify [censored] statements:

All students currently enrolled at CWRU Physics must be treated equally — given the same level of academic support and held to the same standards. No enrolled student is more or less legitimate than any other enrolled student.

As I’ve reminded both [censored] over email, one of the primary purposes of the University is the seeking of truth; and, I’d add here, the dissemination of that knowledge. The blatantly dishonest manner by which I was treated by [censored] and my alma mater speaks to the many points I have already raised above. Specifically, CWRU is an institute of higher learning in the United States, whose constitution enshrines the freedom of expression within its First Amendment. Within Academia, the spirit of open debate should be encouraged and firmly upheld; free expression should certainly not be stifled. By censoring dissident views like mine Case Physics is joining many other increasingly illiberal academic organizations and institutions in betraying their American ideals. Additionally, if faculty believe that students need to be “protected” (really, coddled) just because they are gay or Black or female, etc. — know that Case Physics would then be guilty of the bigotry of lowered expectations. Students should and ought to be expected to defend their ideas and arguments robustly, regardless of their backgrounds or physical characteristics. This is especially so if we are interested in training scientists; namely, scientific results should be robust under intense interrogation. What makes this incident even more egregious is, by endorsing the DIE-based student group and publicly blocking dissidents, Case Physics is protecting a particular brand of politics. This merely adds yet another data point to my hypothesis that our scientific communities are corrupted by ideologies that have little to do with Science itself. It also corroborates my fear that DIE is in fact a religion of Academia; a set of far Left doctrines that we are not allowed to question.

I was challenged by [censored] on why I did not first write to them privately. I hope they have done 2 microseconds of self-reflection to recognize that, through their hyperbolic, unprofessional, and dishonest responses to my deep concerns about DIE eroding meritocracy and scientific standards, they have confirmed my worst fears that doing so would merely have led to soured relationships — and I would have achieved nothing else. (In fact, [censored] implied that I needed to delete all my social media posts criticizing Case Physics before he would engage me: does that sound like he is truly interested in discussion?) It should be clear by now, my public posts on Case were not about anything personal at all; in fact, I even tried to find out from Lydia Kisley on Twitter how much support DIE related policies had among Case faculty, so I could better understand what had really happened.

On the other hand, the fear of speaking out is real. Since making my views public, (ex-)academics have written to me to express their support privately, but I sense many are worried about losing their jobs or facing serious backlash. The two rebuttals to PfJ were written by our professional colleagues; but they too felt the strong need to remain anonymous. On Facebook, when I asked Daniel Harlow, one of the co-authors of PfJ, why he and his collaborators did not even bother replying to the rebuttals; he immediately compared the two to crackpots on the internet. Even here in Taiwan, a physicist based at a neighboring city wrote to my Department Chair last year to try to cancel me for my views on Twitter. I’ve also lost count of how many FB groups, Twitter accounts, youtube commentary, etc. I’ve been blocked / censored / kicked off / etc. by science-related organizations or persons. Hence, by speaking out publicly, I am doing my (tiny) part to push back against the severe corrosion of academic ideals. Let me remind all of you, as academic faculty — i.e., with jobs more secure than most other careers — we have an obligation to speak up and defend the right for all views to be heard and discussed, especially in these highly illiberal times.

Rather importantly, as I’ve also reminded [censored]: the role of a scientist is not to cower to the latest political ideologies, however overwhelmingly fashionable they may be, but to speak the truth as he/she sees it.

[Paragraph removed.]

To speak from my conscience: I believe both moral and scientific authority must be earned, and certainly cannot be forced. As a scientist, I’ve always believed in holding myself to high standards of integrity, as far as my professional conduct is concerned. Merely sitting in meetings and asking the occasional clever question really should not qualify one for authorship on a scientific paper; yet, in theoretical physics, I worry this is a widespread practice due to the dwindling resources/jobs/etc and the dire need to appear productive. As an academic, I do in fact strive hard to mentor students to the best of my abilities. This is in no small part due to the poor standards of mentorship I have personally witnessed or experienced throughout my time as an academic. (Why don’t theoretical physicists discuss more frequently and openly such issues regarding intellectual credit and genuine mentorship, instead of chest-beating about DIE? In evolutionary biology-speak, I believe this amounts to whether or not the signaling is honest.) [Censored]

I will make a final appeal, directed especially to the more senior faculty, because according to recent surveys junior academics are increasingly supportive of far Left Illiberalism. Many of the values I am espousing here overlap a great deal with Enlightenment ones and with Classical Liberalism — particularly the need to view one another as individuals, These values, in turn, heavily influenced the Founding Fathers of America, as reflected in their Declaration of Independence and the US Constitution. Why does it take a Singaporean (me) to remind a bunch of Americans what your foundational values ought to be? Importantly, are senior faculty pushing back at all regarding the far Left corruption of Case Physics? Is anyone challenging the extremely poor leadership of [censored] with regards to Identity Politics? Are you too cowardly, afraid to offend your colleagues, even to defend such important values? Or, are all of you part of the same DIE cult?

The senior faculty at Case should also recognize, the particle astrophysics sector of CWRU Physics would not be what it is today without the leadership of your former Chair Lawrence Krauss. So, if you could care less about listening to this nobody writing from Taiwan, you should at the very least listen to Lawrence, one of the few senior physicists who have been speaking out against identity politics:

https://www.wsj.com/articles/the-new-scientific-method-identity-politics-11620581262
https://quillette.com/2020/07/04/podcast-98-physicist-lawrence-krauss-on-why-identity-politics-should-be-kept-out-of-science/
https://quillette.com/2021/06/02/in-defense-of-the-universal-values-of-science/

I end with the following.

I urge Case faculty to push to abolish all DIE policies and withdraw official support for all DIE student groups; and adhere strictly to meritocracy and to judging our fellow scientists (students, faculty, etc.) as individuals. (The APS has gone Woke; and no physics department should be taking money from it to fund DIE.) As private individuals, students should of course enjoy the freedom of association, and be free to form whatever group they wish. But Case Physics should not be endorsing political movements and corrupting Science.
I also urge Case faculty to set an example for students, to uphold the spirit of free expression and open debate, as it is fundamental to robust discussion. This means students should not be shielded from ideas that make them uncomfortable or even offend them. As I’ve already pointed out above, all these likely require senior faculty to take up a strong leadership role and to stand up firmly for American values — if there are any of you left — because your junior colleagues may have long abandoned them in favor of the Social Justice Religion.
Case Physics should reinstate the GRE requirement for graduate admissions; it was waived last year, I remember. The abolishment of SAT, GRE, etc. requirements throughout US academia is evidence of what I’ve been alluding to: DIE does in fact involve lowering the bar, or removing it altogether, so that more Blacks and Hispanics may be admitted, even if they are under-qualified. The GRE is the one test that all domestic and international applicants would need to take; i.e., it should be the most objective measure of ability, since it provides a common standard, yet the Left dominated Academia is so eager to remove it. Is that not suspicious?
[Censored] asserted that my concerns regarding meritocracy and scientific standards need to be judged based on the “very rich trail of public statements on these matters” I’ve made. I suppose this must have came from the faculty who runs the social media accounts of CWRU Physics? Do we have a Grand Wizard of the Holy Church of the Extreme Left residing in my alma mater? (The data indicates, the higher the administrative ranking the academic is, the more extreme Left his/her politics is likely to be.) Let me reiterate: Case Physics, as a science department, should be apolitical. Whoever runs the Department’s social media accounts should keep his/her politics to him/herself; or otherwise, a replacement is most definitely called for.
If Case Physics stubbornly wishes to continue down this DIE path, at the very least invite external experts to speak on the relevant subject matter. No, Chanda Prescod-Weinstein, Jessica Wade, Brian Nord, etc. are most definitely not experts; particularly when their continued activism and celebrity status require there to be something to complain about in the first place. It is simply not in their self-interests to be honest players in this game of Identity Politics. In my view, it is particularly pressing that academics listen to dissident voices, primarily (though not limited to) those who are not Left-leaning; as well as those who have the relevant (scientific) knowledge especially regarding group differences. Some names I’d recommend include: Gad Saad, John McWhorter, Glenn Loury, Heather Mac Donald, Lee Jussim, Noah Carl, Bo Winegard, Alessandro Strumia, Janice Fiamengo, Heather Heying, Bret Weinstein, Charles Murray, Christina Hoff Sommers, etc.

— Yi-Zen Chu

[StrumiaApr21] https://direct. mit.edu/qss/article/2/1/225/ 99129 / Gender-issues-in- fundamental-physics-A

[Wade] https://www.newscientist.com/article/2181160-it-is-2018-so-why-are-we-still-debating-whether-women-can-do-physics/ and https://twitter.com/jesswade/status/1046334690268008448

[Crackpot1] https://www.journals.uchicago.edu/doi/full/10.1086/704991

[Crackpot2] https://slate.com/technology/2017/08/evolutionary-psychology-is-the-most-obvious-example-of-how-science-is-flawed.html

[PinkerYoutube] https://www.youtube.com/watch?v=qqMBUnaI29U

[APAReport] http://psych.colorado.edu/~carey/pdfFiles/IQ_Neisser2.pdf On p93, “African Americans. The relatively low mean of the distribution of African American intelligence test scores has been discussed for many years. Although studies using different tests and samples yield a range of results, the Black mean is typically about one standard deviation (about 15 points) below that of Whites….”

[Blocked] https://strugglesinphysics.wordpress.com/2021/07/13/ideological-corruption-at-cwru-physics-an-update/

Soon after I sent my email — perhaps unsurprisingly — I received the following reply from a Case Physics faculty:

If your email can be blunt, so can mine: fuck off and never email me again.

Additionally, I also sent a letter to CWRU (intern) Scott Cowen and Provost Ben Vinson. (Unfortunately, I only realized after sending the letter that Scott Cowen has since stepped down.)

Freedom of Expression vs DIE (Diversity, Inclusion, Equity) at CWRU Physics

Dear President Scott Cowen and Provost Ben Vinson,

I obtained my PhD from Case Physics in 2010, and am now Associate Professor of Physics at National Central University, Taiwan.

In early April this year, Case Physics’ co-Chairs Corbin Covault and Glenn Starkman endorsed the formation of a minority student DIE-based group (CWRU PURMS) in the department. The group posted the email endorsement letter on their Twitter account, and explicitly stated they wish to hold “DEI-based” discussions. For a while now, I’ve been concerned about the erosion of academia’s scientific and academic standards in the name of DIE; so despite the risk — I’m fully aware we live in highly politically polarized times — I decided to post my concerns directly on the student group’s thread and also reached out on Twitter to Glenn Starkman (who was my PhD co-advisor). Soon after, I was blocked on Twitter by Glenn, CWRU PURMS, and CWRU Physics.

If these were personal affairs, I would not be reaching out to both of you. But Glenn Starkman is the co-Chair of Case Physics, and the Twitter CWRU Physics page is the official account. As such, I think it is fair to assert, CWRU Physics has behaved in a rather illiberal manner by rapidly censoring a concerned alumnus, just because my views were not aligned with theirs. To be sure, over email, I was accused by [censored] of “attacking students” or “delegitimizing” them; and I was therefore blocked in order to “protect” the students; but I will let you both verify who, if anyone, actually received negativity directed at them personally. Glenn further added that my comments are also to be judged based on the “rich trail” of social media comments I’ve left online; and even implied I need to remove my social media posts criticizing Case Physics before he will engage further.

Ideological Corruption at CWRU Physics: An Update

Incidentally, I also gave via Zoom a colloquium talk last Fall (end October 2020). Despite staying up all night here in Taiwan, so that I may deliver my talk at 4 pm EST, I soon noticed the CWRU Physics’ Twitter account did not advertise my colloquium even though it had advertised those in the weeks before and after mine. In view of what had just transpired, I suspect this omission was done on purpose. If this suspicion does turn out to be true, and given my talk was strictly about physics, is this not unprofessional censorship?

As an alumnus, I am heartened to see that CWRU as an institution has in fact adopted a Chicago-like statement (linked below) affirming that it would uphold and protect freedom of expression. If I may speak from my conscience, however: while I can understand many academics may come from a place of good intentions, I fear DIE is not only eroding meritocracy, it is also eating away Academia’s respect for free speech — and, for the case at hand, the freedom of conscience. My experiences described here adds yet another example to the long list of violations of free expression by American academics and/or their institutions, censoring / canceling / censuring / etc. fellow academics for wrongspeak and wrongthink regarding “Diversity” and Identity Politics related issues. Given the protection of free expression is enshrined in the US constitution — which is what makes America one of the most liberal (and, in my opinion, one of the greatest) of all Western democracies — I am worried Case Physics, like many other institutions, is growing extremely unAmerican. We have forgotten, as academics, that the dedication to seek solutions to difficult problems and the responsibility to seek the truth, requires free thought and free speech.

Click to access Policy-on-Freedom-of-Expression-and-Expressive-Activities-final.pdf

Chicago Statement: University and Faculty Body Support

To be clear, I personally do not care if I am blocked on Twitter by my alma mater. But the issue at hand is much larger: I urge both of you to help restore this foundational academic ideal of the freedom of expression to my alma mater, Cae [sic] Physics. While they do not have the obligation to engage every dissident view, both students and faculty — in their capacities as academics — should certainly not be censoring them. Furthermore, to ensure they mature as academics, students should most definitely not be shielded from ideas and viewpoints they might find deeply offensive. Instead, they have the full right to participate in rigorous debate, to push back with reason and logic.

Yours respectfully,
Yi-Zen Chu

P.S. I have copied this to both the Physics Department, as well as FIRE, an organization that has been fighting the corrosion of free expression on American campuses.