Transverse-Traceless Massless Spin-2 Gravitational Waves Are Acausal

Let X^i be the Cartesian coordinate vector joining one end of a laser interferometer arm to another; and let this interferometer be freely-falling in a weakly curved spacetime

(1)     g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}.

Practically all the pedagogical literature on gravitational physics tell us the distortion \delta X^i of this arm due to the presence of a gravitational wave is proportional to the transverse-traceless part of the metric perurbation h_{\mu\nu}:

(2)     \delta X^i = \frac{1}{2} h_{ij}^{\text{TT}} X^j .

But, what does “transverse-traceless” (TT) actually mean here? The field theorist reader would likely think that the h_{ij}^{\text{TT}} must be the gauge-invariant massless spin-2 graviton, which obeys

(2′)     \partial_i h_{ij}^{\text{TT}} = 0 \qquad \qquad \text{(Transverse)}

and

(2”)     \delta^{ij} h_{ij}^{\text{TT}} = 0 \qquad \qquad \text{(Trace-less)} .

Massless Spin-2     The helicity–2 character is the result of these TT conditions; for, each Fourier \vec{k}-mode, it is always possible to find a basis of polarization tensors \epsilon_{ij}^\pm, namely

(2”.I)     h_{ij}^{\text{TT}} = \int \frac{d^3 \vec{k}}{(2\pi)^3} \left\{ \left( a_+[\vec{k}] \epsilon^+_{ij}[\vec{k}] e^{-i |\vec{k}|t} + a_-[\vec{k}] \epsilon^-_{ij}[\vec{k}] e^{-i |\vec{k}|t} \right) e^{i\vec{k}\cdot\vec{x}} + \text{c.c.} \right\} ;

such that under a rotation along the axis \vec{k} through an angle \theta,

(2”’)     \epsilon_{ij}^\pm e^{-i |\vec{k}|t+i\vec{k}\cdot\vec{x}} \to e^{(\pm 2) i \theta} \epsilon_{ij}^\pm e^{-i |\vec{k}|t+i\vec{k}\cdot\vec{x}}.

The \pm 2 may be viewed as the eigenvalues of the generator of rotation on the plane perpendicular to \vec{k}.

Gauge-Invariance     Next, by gauge-invariance, I mean here that, under an infinitesimal change in coordinates

(3)     x^\alpha \to x^\alpha + \xi^\alpha ,

the TT character of this gravitational wave ensures it remains unaltered:

(3′)     h_{ij}^{\text{TT}} \to h_{ij}^{\text{TT}} .

Now, this gauge-invariance is often invoked as a criterion for physical observability: for, if some observable is expressed in terms of the gauge dependent components h_{\mu\nu} in eq. (1), how does one know if the physical effect at hand cannot be rendered trivial simply by choosing an infinitesimally different coordinate system? However, the main point of this post is — the converse most certainly does not hold:

Gauge invariance does not imply physical observability.

The reason is simple: even though h_{ij}^{\text{TT}} is gauge-invariant, it is acausal. More specifically, within the linearized approximation of General Relativity, this massless spin-2 gravitational wave (GW) admits the solution

(4)     h_{ij}^{\text{TT}}[x] = \int_{\mathbb{R}^{3,1}} d^4 x' G_{ij a'b'}[x-x'] T_{a'b'}[x'];

where G_{ij a'b'} is the Green’s function of the TT GW and T_{ab} is the stress-energy tensor. Through a direct calculation, in arXiv: 1902.03294, Yen-Wei Liu and I showed that G_{ij a'b'}[x,x'] is non-zero outside the past light cone of the observer at x. In other words, the signal h_{ij}^{\text{TT}} receives contributions from portions of T_{a'b'} that are spacelike separated from the observer — and therefore cannot be a standalone observable.


Tidal Forces & GW Strain    So, what is one to make of the formula in eq. (2) then? To this end, we first recall that — if \ell^\mu describes the displacement between a pair of infinitesimally nearby timelike geodesics (a pair of freely-falling test masses, for instance); the fully covariant acceleration a^\mu of this displacement vector is driven by the Riemann tensor:

(5)     a^\mu = -R^\mu_{\phantom{\mu} \nu \alpha\beta} U^\nu \ell^\alpha U^\beta.

(The U^\alpha is the unit norm timelike vector tangent to one of the two geodesics.) In a flat spacetime, the Riemann tensor is exactly zero; i.e., a pair of parallel lines will remain parallel because their relative acceleration is zero. Now, at first order in the perturbation h_{\mu\nu}, both sides of eq. (5) must be gauge-invariant since their ‘background value’ (evaluated on g_{\mu\nu} = \eta_{\mu\nu}) is zero. This in turn means we can choose any gauge we wish. Synchronous gauge, where the perturbations are strictly spatial

(5′)     h_{\mu\nu} d x^\mu d x^\nu \to h_{ij}^{\text{(s)}} d x^i d x^j ,

is particularly pertinent in this context of 2 infinitesimally close-by free falling test masses. For, if the x^0 of the synchronous-gauge coordinate system refers to the proper times of these free falling objects, their spatial coordinates are then automatically time independent, and

(5”)     U^\mu = \delta^\mu_0 .

If we assume the clocks on this pair of test masses are synchronized at some initial time t_0, then one may demonstrate using eq. (5) they will continue to remain so for later times; namely, a^0[x^0 > t_0] = 0 if \ell^0[x^0 = t_0] = 0 = U^\sigma \nabla_\sigma \ell^0[x^0 = t_0]. Employing eq. (5”), the spatial tidal forces described by the geometrically induced relative acceleration is now

(6)     \delta_1 a^i = \delta_1 R_{0i0j} \ell^j + \mathcal{O}[h^2] ;

with the notation \delta_1 R_{0i0j} denoting the 0i0j components of the linearized Riemann tensor.

Within the synchronous gauge, the proper distance between two free falling test masses \vec{Z}_1 and \vec{Z}_2 at a given time t (accurate to first order in perturbations) is

(7)     L_{1 \leftrightarrow 2}[t] = R \left( 1 - \frac{1}{2} \widehat{R}^i \widehat{R}^j \int_0^1 h_{ij}^{\text{(s)}}\left[ t, \vec{Z}_1 + \lambda(\vec{Z}_2-\vec{Z}_1) \right] d \lambda + \mathcal{O}[h^2] \right) , \\ R \equiv |\vec{Z}_1 - \vec{Z}_2|, \qquad \widehat{R}^i = (\vec{Z}_1 - \vec{Z}_2)/R ;

from which, we see that the fractional distortion \delta L[t]/R is

(7′)     \delta L[t]/R = - \frac{1}{2} \widehat{R}^i \widehat{R}^j \int_0^1 h_{ij}^{\text{(s)}}\left[ t, \vec{Z}_1 + \lambda(\vec{Z}_2-\vec{Z}_1) \right] d \lambda + \mathcal{O}[h^2].

Moreover, the linearized Riemann \delta_1 R_{0i0j} in the synchronous gauge reads

(8)     \delta_1 R_{0i0j} = -\frac{1}{2} \partial_0^2 h_{ij}^{\text{(s)}}.

Remember the linearized Riemann is gauge-invariant, so it ought to be possible to re-express \delta_1 R_{0i0j} in terms of gauge-invariant metric perturbation variables. (More on this below.) In fact, what Yen-Wei and I argued in arXiv:1902.03294 was that, in the far zone where (observer-source distance)/(characteristic timescale of source)\gg 1,

(8′)     \delta_1 R_{0i0j} = -\frac{1}{2} \partial_0^2 h_{ij}^{\text{TT}} \qquad \text{(Far zone)}.

Therefore, in frequency space

(8”)     \widetilde{h}_{ij}^{\text{TT}}[\omega,\vec{x}] = \int_{\mathbb{R}} d t e^{i \omega t} h_{ij}^{\text{TT}}[t,\vec{x}] ,

that the linearized Riemann is gauge-invariant allows us to equate (8) and (8′) to conclude — for finite frequencies \omega

(9)     \widetilde{h}_{ij}^{\text{TT}}[\omega,\vec{x}] = \widetilde{h}_{ij}^{\text{(s)}}[\omega,\vec{x}]  \qquad \text{(Far zone)} .

Important aside     By placing \vec{0} at the center-of-mass of the material source of gravity, in the same far-zone limit, the transverse-traceless GW h_{ij}^{\text{TT}} reduces to

(9′)     h_{ij}^{\text{TT}} \approx \left( P_{ia} P_{jb} - \frac{1}{2} P_{ij} P_{ab} \right) h_{ab}[\text{de Donder}] , \\ P_{ij} = \delta_{ij} - \widehat{r}_i \widehat{r}_j ,  \qquad \widehat{r}_i \equiv x_i/|\vec{x}| , \qquad \widehat{r}_i P_{ij} = 0 .

Namely, the far-zone massless spin-2 GW is the de Donder gauge gravitational perturbation projected locally-in-space transverse to the propagation direction. But the de Donder gauge graviton is in fact causally dependent on the stress tensor; in the far zone, in particular,

(9”)     h_{ij}[\text{de Donder}] \approx -\frac{4 G_{\text{N}}}{r} \int_{\mathbb{R}^3} d^3 \vec{x}' T_{ij}[t-r+\vec{x}'\cdot\widehat{r},\vec{x}'] . \qquad r \equiv |\vec{x}| .

This means the far zone TT GW in eq. (9′) is causal, even though its full form in eq. (4) is not. The reason is, the acausal portions begin at higher order in 1/r. In the GW literature, the local-in-space projection in eq. (9′) — what Ashtekar and Bonga, referenced below, dubbed h_{ij}^{\text{tt}} to distinguish it from h_{ij}^{\text{TT}} in equations (2′) and (2”) — is actually the one that is employed, not the tranverse-traceless one subject to equations (2′) and (2”). We see, the reason why it is possible to get away with mixing these two distinct notions of transverse-traceless projections is that they coincide when \omega r \gg 1; i.e., in the far zone. (Note: Racz and Ashtekar-Bonga, whose papers can be found below, have correctly complained that the GW literature wrongly mixes ‘tt’ versus ‘TT’.)

Summary     Let us sum up the discussion within this section. In the far zone, the fractional distortion of the proper distance between the pair of free-falling test masses \vec{Z}_1 and \vec{Z}_2 at a given time t is

(10)     \frac{\delta L[t]}{R} = - \frac{1}{2} \widehat{R}^i \widehat{R}^j \int_0^1 h_{ij}^{\text{TT}}\left[ t, \vec{Z}_1 + \lambda(\vec{Z}_2-\vec{Z}_1) \right] d \lambda + \mathcal{O}[h^2].

This formula is to be understood as valid only for finite frequencies — for instance, LIGO is built to be sensitive to a limited bandwidth centered roughly at 100Hz. Otherwise, equating (8) and (8′), which was what led to eq. (9)-(10), actually misses the initial h_{ij}^{\text{(s)}} and its time derivative; in frequency space these initial conditions correspond to zero-\omega Dirac \delta-function terms. In the limit where the wavelength of the GW is long compared to R, so h_{ij}^{\text{TT}} is approximately constant between \vec{Z}_1 and \vec{Z}_2, eq. (10) then reduces to

(10′)     \delta L[t]/R \approx - \frac{1}{2} \widehat{R}^i \widehat{R}^j h_{ij}^{\text{TT}} + \mathcal{O}[h^2].

This is equivalent to eq. (1); but to arrive at it we have assumed the following.

  • The GW detector is in the far zone.
  • The GW detector is only sensitive to finite gravitational wave frequencies.
  • The GW detector’s proper size is much smaller than the gravitational wavelength.

Dynamical Degrees-Of-Freedom vs. Physical Observables     In field theory speak, one often hears the statement that “4D Einstein-Hilbert gravity has only 2 dynamical degrees-of-freedom”. In its linearized form, we shall see this statement amounts to:

Of all the gauge-invariant variables formed from the metric perturbation h_{\mu\nu} in eq. (1) — the transverse-traceless tensor h_{ij}^{\text{TT}} = D_{ij}; the transverse vector V_i; and the scalars \Psi and \Phi — only the tensor obeys a wave equation.

To build h_{ij}^{\text{TT}} = D_{ij}, V_i, \Psi and \Phi out of the perturbation h_{\mu\nu}, refer to equations (A10), (A15) and (A16) of arXiv: 1611.00018. (Put d=4; remove the over-bars and note that h_{\mu\nu}[\text{here}] = \chi_{\mu\nu}[\text{1611.00018}].) What I wish to highlight here are the (3+1)D version of the equations-of-motion in (A25) and (A26):

(11)     \vec{\nabla}^2 \Phi = 8\pi G_{\text{N}} \rho, \qquad \Phi - \Psi = 16\pi G_{\text{N}} \Upsilon, \\ \vec{\nabla}^2 V_i = 16 \pi G_{\text{N}} \Sigma_i, \qquad \partial^2 h_{ij}^{\text{TT}} = -16\pi G_{\text{N}} \sigma_{ij}.

The transverse-traceless conditions of equations (2′) and (2”) tell us, of the 3+(3^2-3)/2=6 components of h_{ij}^{\text{TT}}, only 6-3-1=2 are independent. However despite this “2 d.o.fs” assertion regarding the TT GW, as I have already pointed out above its solution is acausal and cannot possibly be a standalone physical observable. In eq. (11) the \sigma_{ij} is in fact a non-local functional of the spatial components of the stress tensor — heuristically, T_{ij} is smeared out over all space in such a manner that the resulting object \sigma_{ij} obeys the constraints \partial_i \sigma_{ij} = 0 = \sigma_{ij} \delta^{ij}.

What, then, is one to make of this acausality; as well as the gauge-invariant content of linearized gravitation? A partial answer is offered by the spatial tidal forces exerted by geometric curvature, encoded within the \delta_1 R_{0i0j} discussed above. Yen-Wei and I showed that, even though the TT GW h_{ij}^{\text{TT}} and its acceleration \partial_0^2 h_{ij}^{\text{TT}} are acausal, the vector V_i and scalars \Phi and \Psi appear in \delta_1 R_{0i0j} in such a way to precisely cancel out the acausal contributions from the tensor; with the end result yielding tidal forces that are strictly causally dependent on the material stress tensor:

(12)     \delta_1 R_{0i0j} = \frac{1}{2} \left( -\frac{1}{2} \partial_i \partial_j \Psi - \delta_{ij} \partial_0^2 \Phi + \partial_0 \partial_{\{ i} V_{j \}} - \partial_0^2 h_{ij}^{\text{TT}} \right) = - \frac{1}{2} \left( \partial_0^2 h_{ij}^{\text{TT}} \right)_{\text{causal}}.

Therefore, the tidal squeezing and stretching of a Weber bar or of a laser interferometer’s arms is not to be attributed to the entire spin-2 massless graviton — because of its acausal character — but to only the causal part of its acceleration. Even in the (quasi-)static limit where the \Phi, V_i, h_{ij}^{\text{TT}} in eq. (12) all appear to become negligible; say, for instance, the contribution to the tides on Earth due to differential gravitational tugs from either the Moon or the Sun; we should not attribute the rising and ebbing of the oceans to the second derivatives of the Newtonian-like potential \Psi in eq. (11). Rather, on grounds that physical tidal forces ought to be causal, according to eq. (12), it still has to be attributed to the causal part of the TT tensor perturbation’s acceleration.


Micro-causality in QFT     If you have taken a course on Quantum Field Theory, you might have been told that the amplitude for a particle to propagate from y to x is given by the vacuum expectation value \langle 0 \vert \varphi[x] \varphi[y] \vert 0 \rangle (for scalar particles \varphi). However, a direct calculation for non-interacting scalars would reveal this object is non-zero for spacelike separated x and y; i.e., a particle has a non-zero quantum mechanical amplitude to propagate outside the light cone. See discussion in \S 2.4 of Peskin and Schroeder (P&S) for instance. P&S goes on to assert

To really discuss causality, however, we should ask not whether particles can propagate over spacelike intervals, but whether a measurement performed at one point can affect a measurement at another point whose separation from the first is spacelike. The simplest thing we could try to measure is the field \phi(x), so we should compute the commutator [\phi(x),\phi(y)]; if this commutator vanishes, one measurement cannot affect the other. In fact, if the commutator vanishes for (x-y)^2 < 0, causality is preserved quite generally, … [truncated] — Chapter 2, page 28

At the quantum level, are the transverse massless spin-1 photon A_i^\text{T} (subject to \partial_i A_i^{\text{T}} = 0) or spin-2 graviton field h_{ij}^{\text{TT}} physically observable? That is, can we perform a direct measurement on them? P&S does not tell us, but although the commutators of free scalar fields vanish outside the light cone — they obey micro-causality — the helicity-1 and -2 photons and gravitons do not.

(13)     \left[ A_i^{\text{T}} [x], A_j^{\text{T}} [y] \right] \neq 0, \qquad \qquad (x-y)^2 < 0 ;

(13′)     \left[ h_{ij}^{\text{TT}}[x], h_{ab}^{\text{TT}}[y] \right] \neq 0, \qquad \qquad (x-y)^2 < 0 .

This is simply because their commutators are proportional to the difference between their corresponding retarded and advanced Green’s functions. As already alluded to after eq. (4), these retarded/advanced transverse Green’s functions are in fact non-zero outside the light cone. I end with the following question:

Can this violation of micro-causality by massless spin-1 photons be exploited within a physical setup?

Note added: I forgot to mention an interesting related discussion that took place over at Distler’s blog regarding micro-causality. My sense is, he knows a whole lot more than I do — but, sadly, he has closed his comments section for the post.


References

  • I. Racz, “Gravitational radiation and isotropic change of the spatial geometry,” arXiv:0912.0128 [gr-qc]
  • A. Ashtekar and B. Bonga, “On the ambiguity in the notion of transverse traceless modes of gravitational waves,” Gen. Rel. Grav. 49, no. 9, 122 (2017) doi:10.1007/s10714-017-2290-z [arXiv:1707.09914 [gr-qc]]
  • A. Ashtekar and B. Bonga, “On a basic conceptual confusion in gravitational radiation theory,” Class. Quant. Grav. 34, no. 20, 20LT01 (2017) doi:10.1088/1361-6382/aa88e2 [arXiv:1707.07729 [gr-qc]]
  • Y. Z. Chu and Y. W. Liu, “The Transverse-Traceless Spin-2 Gravitational Wave Cannot Be A Standalone Observable Because It Is Acausal,” arXiv:1902.03294 [gr-qc].
  • Y. Z. Chu, “More On Cosmological Gravitational Waves And Their Memories,”
    Class. Quant. Grav. 34, no. 19, 194001 (2017) doi:10.1088/1361-6382/aa8392
    [arXiv:1611.00018 [gr-qc]].
  • S. Weinberg, “Photons and gravitons in perturbation theory: Derivation of Maxwell’s and Einstein’s equations,” Phys. Rev. 138, B988 (1965).
    doi:10.1103/PhysRev.138.B988

Linear Displacement Gravitational Wave Memory

There has been a recent surge of interest in the phenomenon of gravitational memory, likely due to investigations undertaken by Andrew Strominger’s group at Harvard — see here for a pedagogical treatment — linking memory to symmetries and their corresponding Ward identities in the “soft limit” (i.e., where the gravitational signals’ frequencies are low/wavelengths are long). Gravitational memory itself was first discovered (as I understand it) by Zel’dovich and Polnarev: there is a permanent distortion of space due to stars scattering off each other on unbound trajectories.

I first stumbled upon this phenomenon myself while examining the causal structure of gravitational waves, namely how they propagate both on and within the null cone, in cosmological spacetimes. I found that the portion of gravitational waves (GWs) that travel inside the light cone (aka its “tail”) does not always decay with increasing distance from its source. This leads to a novel, albeit tiny, tail-induced gravitational memory that has no counterpart in the flat spacetime limit.

In this post, I will focus on linear displacement gravitational memory. As we shall see, this is the permanent distortion of space due to the passage of a primary GW train, induced directly by the matter source itself. As discovered by Christodoulou and independently by Blanchet and Damour, there is also a contribution to GW memory from the stress-energy of the GWs themselves, which is dubbed “nonlinear memory”; I hope to discuss this in a later post.

Synchronous gauge     To this end, we shall work in the synchronous gauge, because it allows us to readily discuss the proper geodesic length between two freely-falling observers at a given time. In particular, synchronous gauge refers to the coordinate system where the metric has no time-time and time-space components, namely,

(1):     ds^2 = dt^2 + g_{ij} dx^i dx^j .

The interpretation is that spacetime is foliated by the worldlines of free falling timelike trajectories (i.e., spatial point-like “observers”), with proper time t and spatial trajectories \vec{x}. This interpretation may be confirmed by verifying, co-moving timelike geodesics Z^\mu that have time independent spatial components:

(2):     Z^\mu = (t, Z^i), \qquad\qquad Z^i = \text{constant}

in fact satisfy the geodesic equation automatically.

What Is Displacement Gravitational Wave Memory?     Consider a pair of test masses (Z_1,Z_2) co-moving in a weakly curved spacetime,

(3):     ds^2 = dt^2 - (\delta_{ij} - h_{ij}) dx^i dx^j , \qquad \qquad|h_{ij}| \ll 1.

As we have discussed in a previous post, we may use Synge’s world function — which also defines the action for affinely-parametrized geodesics — to express the proper geodesic spatial distance L[t] between \vec{Z}_1 and \vec{Z}_2 at time t.

(4):     L[t] \approx R \left( 1 - \frac{1}{2} \widehat{R}^i \widehat{R}^j \int_0^1 h_{ij}\left[t, \vec{Z}_1 + \lambda (\vec{Z}_2-\vec{Z}_1)\right] d\lambda + \mathcal{O}[h^2] \right) ;

where

(5):     \widehat{R}^i \equiv \frac{\vec{Z}_1 - \vec{Z}_2}{R}, \qquad \qquad R \equiv |\vec{Z}_1-\vec{Z}_2| .

Therefore, there is a permanent distortion \Delta L \equiv L[t \to +\infty] - L[t \to -\infty], i.e., gravitational memory, if there is a non-trivial “DC-shift” of the gravitational perturbation between the two test masses over a large time period enveloping the duration of the primary GW train. The fractional distortion, in particular, is

(6):     \delta L[t]/R = - \frac{1}{2} \widehat{R}^i \widehat{R}^j \int_0^1 \Delta h_{ij}\left[\vec{Z}_1 + \lambda (\vec{Z}_2-\vec{Z}_1)\right] d\lambda ,

where

(6′):     \Delta h_{ij}\left[\vec{z}\right] \equiv h_{ij}\left[t \to +\infty, \vec{z}\right] - h_{ij}\left[t \to -\infty, \vec{z}\right] .

By setting up pairs of test masses with different orientations \widehat{R}^i, one can probe the full pattern of GW memory encoded within \Delta h_{ij}. In other words, the distortion of space is generically anisotropic.

Linear GW memory     Linear memory arises directly due to the matter source itself. Let r be the spatial distance between the observer and the center-of-mass of the said GW source. At finite frequencies \{ \omega \}, the synchronous gauge metric perturbation in the far zone |\omega|r \gg 1 and at first order in G_\text{N} reads

(7):     h_{ij} = h^{tt}_{ij} = -\left( P_{ia} P_{jb} - \frac{1}{2} P_{ij} P_{ab} \right) \frac{4 G_{\text{N}}}{r} \int_{\mathbb{R}^3} d^3 \vec{x}' T_{ab}[t-r+\vec{x}'\cdot\vec{x}/r, \vec{x}'] ,

with the projector

(7′):     P_{ij} \equiv \delta_{ij} - \widehat{r}_i \widehat{r}_i, \qquad \widehat{r}^i \equiv x^i/r.

Here, T_{ij} are the spatial components of the matter stress energy tensor. (Nonlinear memory would involve that of the GWs themselves.) Because t-r+\vec{x}'\cdot\vec{x}/r is essentially the retarded time (up to relativistic corrections), what eq. (7) inserted into eq. (6) teaches us is that:

Since linear GWs propagate on the null cone in 4D Minkowski, the corresponding memory is really a probe of the difference in the asymptotic — i.e., far future versus far past — configurations of the matter source itself.

Causal Structure
Gravitational memory measured by the GW detector (world-line on the left) is the difference between the gravitational perturbation at C and that at A; for linear memory, this in turn probes the difference between the matter configuration (world-tube on the right) at C’ and that at A’. The dashed segment on the matter world-tube is the dominant duration of gravitational radiation production. Figure from arXiv: 1611.00018.

Cosmology     In a spatially flat FLRW spacetime, namely

(8):     g_{\mu\nu} = a[\eta]^2 \eta_{\mu\nu} , \qquad x^\mu \equiv (\eta,\vec{x}) .

I was able to show in arXiv:1504.06337 that the null cone portion of the massless scalar Green’s function takes a universal form, as the Minkowski Green’s function divided by 1 power of the scale factor each at the observer and emission time:

(9):     G^{\text{(light cone)}}_{\text{4D FLRW}}[x,x'] = \frac{\delta\left[ \eta-\eta' - |\vec{x}-\vec{x}'| \right]}{4\pi a[\eta] a[\eta'] |\vec{x}-\vec{x}'|} .

Now, in spatially flat cosmologies driven by perfect fluids, GWs obey a massless scalar wave equation. I also estimated that, for the most part, GW tails in cosmology are highly suppressed unless the time-duration of and observer distance to the source are of cosmological time/length scales. Altogether, these imply the GW memories known in 4D Minkowski spacetime should carry over to spatially flat FLRW, except for the redshift (from the 1/a[\eta]) due to cosmic expansion.


References

  • Zel’Dovich Y B and Polnarev A G 1974 Astron. Zh. 51 30 [Sov. Astron. 18, 17 (1974)]
  • A. Strominger, “Lectures on the Infrared Structure of Gravity and Gauge Theory,” arXiv:1703.05448 [hep-th].
  • D. Christodoulou, “Nonlinear nature of gravitation and gravitational-wave experiments,” Phys. Rev. Lett. 67, 1486 (1991)
  • Blanchet and Damour, 1989.
  • Y.Z. Chu, “Transverse traceless gravitational waves in a spatially flat FLRW universe: Causal structure from dimensional reduction,” Phys. Rev. D 92, no. 12, 124038 (2015) doi:10.1103/PhysRevD.92.124038 [arXiv:1504.06337 [gr-qc]].

 

Schwinger-Keldysh Action Principles for the Damped SHO & 4D Majorana Fermion

Motivation     How does one write down an action for the damped harmonic oscillator \vec{x}? Denoting each time derivative as an overdot,

(1):    m \ddot{\vec{x}} + f \dot{\vec{x}} + \omega^2 \vec{x} = 0 ,

where m is the mass, f is friction, and \omega is the oscillation frequency of the particle in the limit of zero friction.

More generally, how does one write down an action, not necessarily for particle mechanics, that does not require specifying boundary values?

It turns out that these questions are intimately related to the Schwinger-Keldysh formalism behind the computation of expectation values of quantum operators, as well as the treatment of out-of-equilibrium and/or open quantum systems. Here, I will merely focus on the (semi-)classical limit of two specific examples.

General Strategy     The strategy goes as follows. First double the number of degrees of freedom. For example, if \vec{x} is the trajectory of the simple harmonic oscillator particle, we would now have \vec{x}_1 and \vec{x}_2. The full action then takes the form

(2):    S_{\text{total}} \equiv S_0[\vec{x}_1,\dot{\vec{x}}_1] - S_0[\vec{x}_2,\dot{\vec{x}}_2] + S_{\text{IF}}[\vec{x}_1,\dot{\vec{x}}_1; \vec{x}_2,\dot{\vec{x}}_2 ] ,

where the two S_0 are the same except one is evaluated on \vec{x}_1 and the other on \vec{x}_2; while the “influence action” S_{\text{IF}} couples the \vec{x}_1 and \vec{x}_2 but has to obey the anti-symmetry property:

(2′):    S_{\text{IF}}[ \vec{x}_1,\dot{\vec{x}}_1; \vec{x}_2,\dot{\vec{x}}_2 ] = -S_{\text{IF}}[ \vec{x}_2,\dot{\vec{x}}_2; \vec{x}_1,\dot{\vec{x}}_1 ] .

The difference between the action S_0 evaluated on copy-1 and that on copy-2 in eq. (2) arises, within the quantum context, from the Schwinger-Keldysh path integral when describing the time evolution of the density matrix, which plays a key role in the computation of expectation values of quantum field operators. Additionally, the influence action in eq. (2′) can be argued to arise from “integrating out” degrees of freedom.

The full action involves integrating the degrees of freedom from some initial time t_i to the final time t_f and — if fields (as opposed to particles) are involved — over the appropriate spatial domain. However, instead of the usual boundary values, one now requires that the copy-“1” and copy-“2” of the degrees of freedom to be specified at the initial time t_i. At the final time t_f we do not fix their trajectories but merely demand that the two copies coincide there:

(3):    \vec{x}_1[t_f] = \vec{x}_2[t_f]; \qquad\qquad \dot{\vec{x}}_1[t_f] = \dot{\vec{x}}_2[t_f] .

This necessarily means their variation must also coincide:

(3′):    \delta\vec{x}_1[t_f] = \delta\vec{x}_2[t_f]; \qquad\qquad \delta\dot{\vec{x}}_1[t_f] = \delta\dot{\vec{x}}_2[t_f] .

With these conditions in mind, we then demand that the total action S_{\text{total}} be stationary under the variation of both copies of the degrees of freedom. Only after the ensuing equations-of-motion are obtained, do we set the two copies to be equal.


Damped SHO     Let us now proceed to show that the damped harmonic oscillator of eq. (1) follows from the action

(4):    S_{\text{DSHO}} \equiv \int_{t_i}^{t_f} d t \left\{ \left(\frac{1}{2} m \dot{\vec{x}}_1^2 -  \frac{1}{2} \omega^2 \vec{x}_1^2\right) - \left(\frac{1}{2} m \dot{\vec{x}}_2^2 - \frac{1}{2} \omega^2 \vec{x}_2^2\right) - \frac{f}{2} (\vec{x}_1 - \vec{x}_2)\cdot(\dot{\vec{x}}_1 + \dot{\vec{x}}_2) \right\} .

Demanding the action S_{\text{DSHO}} be stationary under variation with respect to both \vec{x}_1 and \vec{x}_2,

(5):    0 = \delta S_{\text{DSHO}} \\ = \int_{t_i}^{t_f} d t \left\{ \delta\vec{x}_1 \cdot \left( -m \ddot{\vec{x}}_1^2 - \omega^2 \vec{x}_1 - \frac{f}{2} (\dot{\vec{x}}_1 + \dot{\vec{x}}_2) + \frac{f}{2} (\dot{\vec{x}}_1 - \dot{\vec{x}}_2) \right) - (1 \leftrightarrow 2) \right\} + \text{BT};

with the boundary terms

(5′):    \text{BT} = \Big[ m (\delta\vec{x}_1\cdot\dot{\vec{x}}_1 - \delta\vec{x}_2\cdot \dot{\vec{x}}_2) - \frac{f}{2} (\vec{x}_1 - \vec{x}_2)\cdot (\delta\vec{x}_1+\delta\vec{x}_2) \Big]_{t_i}^{t_f} .

Remember, from eq. (3), that we fix the initial conditions \delta \vec{x}_1[t_i] = 0 = \delta \vec{x}_2[t_i]; this sets to zero all the terms in the lower limit. Whereas, for the upper limit, we are to set \vec{x}_1[t_f] = \vec{x}_2[t_f]; \delta \vec{x}_1[t_f] = \delta \vec{x}_2[t_f]; \dot{\vec{x}}_1[t_f] = \dot{\vec{x}}_2[t_f] as well as \delta \dot{\vec{x}}_1[t_f] = \delta \dot{\vec{x}}_2[t_f]; and only then does it vanish.

With the boundary terms vanishing, the principle of stationary action then yields the two independent equations

(5”):     m \ddot{\vec{x}}_1 + f \ \dot{\vec{x}}_2 + \omega^2 \vec{x}_1 = 0

and

(5”’):    m \ddot{\vec{x}}_2 + f \ \dot{\vec{x}}_1 + \omega^2 \vec{x}_2 = 0 .

Setting \vec{x}_1 = \vec{x}_2 in equations (5”) and (5”’) then returns the DSHO equation of eq. (1).


4D Majorana Fermion     For the second example, let us turn to the Majorana fermion, which unlike its Dirac cousin, only requires either the chiral left or chiral right SL[2,\mathbb{C}] spinor — but not both. One such version is provided by the equation

(6):     i \bar{\sigma}^\mu \partial_\mu \psi = m \epsilon \cdot \psi^* ,

where \psi is a 2-component spinor, \bar{\sigma}^0 is the 2 \times 2 identity matrix; \bar{\sigma}^i \equiv -\sigma^i with \{\sigma^i\} being the Hermitian Pauli matrices; m is the fermion’s mass, and \epsilon is the 2D Levi-Civita tensor. At the semi-classical level, and at first sight, you might think that the right hand side of eq. (6) could arise from a Lagrangian density of the form

(6′):     \mathcal{L}_{\text{Majorana mass}} = -\frac{m}{2} \left( \psi^\dagger \epsilon \psi^* + \psi^{\text{T}} \epsilon^\dagger \psi \right)

But upon closer examination you’d discover this Lagrangian density is identically zero*, as the Levi-Civita tensor is anti-symmetric and therefore

(6”):     \psi^\dagger \epsilon \psi^* = 0 = \psi^{\text{T}} \epsilon^\dagger \psi .

But as it turns out, the doubled-field formalism allows one to write down a Lagrangian density. It is given by

(7):     S_{\text{Majorana}} = \int_{t_i}^{t_f} dt \left(  \psi_1^\dagger i \bar{\sigma}^\mu \partial_\mu \psi_1 - \psi_2^\dagger i \bar{\sigma}^\mu \partial_\mu \psi_2  + \mathcal{L}_{\text{IF Majorana Mass}} \right)  ,

where the Majorana mass term is now part of the `influence Lagrangian’  that couples the two copies:

(7′):     \mathcal{L}_{\text{IF Majorana Mass}} = - \frac{m}{2} \left( \psi_1^\dagger \epsilon \psi_2^* + \psi_1^{\text{T}} \epsilon^\dagger \psi_2 \right) .

Notice the terms in eq. (7′) are similar to those in eq. (6′) but they do not vanish despite the anti-symmetric nature of \epsilon, because we now have two distinct copies of the spinor field.

A similar variational calculation to the one performed for the DSHO would yield eq. (6) from the action in eq. (7). The primary difference from the DHO is that, fermionic systems are first order ones, and therefore only the two copies of the fields — but not their derivatives — need to match at t_f, to ensure the boundary terms (analogous to the ones in eq. (5′)) vanishes.

I don’t yet know of any potential physical applications of such a perspective. What sort of open quantum systems would yield eq. (7′)?


* Upon quantization — as the referee of my paper below correctly emphasized — these Majorana fermion fields would still obey anti-commutation relations and, hence, Fermi-Dirac statistics. In fact, this is usually how the Majorana mass Lagrangian in eq. (6′) is justified: unlike the case of the Dirac mass terms, one has to introduce Grassmannian variables from the outset, so that eq. (6”) is no longer true.


Remark added     Before submitting to and getting the paper (arXiv:1708.00338) published in JHEP, I had actually attempted to submit it to a different journal. Unfortunately, it was rejected by the editor, without him putting my paper through the peer-review process. I wrote to the editor directly and have not received any reply to date. Is this considered scientific/ethical behavior? What is to prevent journal editors from doing this to non-mainstream type of scientific work such as mine? On the other hand, the JHEP referee did pose some legitimate objections to my presentation, but s/he nonetheless found the work “interesting”.


References

  • J. Schwinger, “Brownian Motion of a Quantum Oscillator,” J. Math. Phys. 2, 407 (1961).
  • L. V. Keldysh, Zh. Eksp. Teor. Fiz. 47, 1515 (1964), [English translation, Sov. Phys. JEPT 20, 1018 (1965)].
  • R. D. Jordan, “Effective field equations for expectation values,” Phys. Rev. D33, 444 (1986).
  • C. R. Galley, D. Tsang and L. C. Stein, “The principle of stationary nonconservative action for classical mechanics and field theories,” arXiv:1412.3082 [math-ph].
  • C. R. Galley, “Classical Mechanics of Nonconservative Systems,” Phys. Rev. Lett. 110, no. 17, 174301 (2013) doi:10.1103/PhysRevLett.110.174301 [arXiv:1210.2745 [gr-qc]].
  • J. Polonyi, “Environment Induced Time Arrow,” arXiv:1206.5781 [hep-th].
  • Y.-Z. Chu, “A Semi-Classical Schwinger-Keldysh Re-interpretation Of The 4D Majorana Fermion Mass Term,” J. High Energ. Phys., (2018) 2018: 13; arXiv:1708.00338 [hep-th].

Gender Differences for the Lazy/Busy

When I was a teenager (or younger?) I had already read that there are brain differences between men and women. Unfortunately, having spent nearly 2 decades in the US, the information I received regarding gender differences was often muddled; and only later on I began to realize, this was probably due to ideology on the Left. I wish to report that this unscientific behavior can be found throughout Physics and Astrophysics, driven by Leftist politics and radical feminism. That the noble demand for equal rights for men and women does not imply nor require that the genders have to be the same in every aspect — this is clearly not properly appreciated by many in Academia. Unfortunately, much of the hypersensitivity to gender issues is driven by the unfounded desire to see equal representation of women and men in physics, instead of allowing them free rein to choose their careers and judging people purely by merit, as the Scientific Method requires.

Recently, particle theorist Alessandro Strumia gave a talk at CERN’s 1st Workshop on High Energy Theory and Gender. The main thrust of his talk was to challenge the mainstream narrative that High Energy Theoretical Physics has much fewer women than men because of rampant discrimination. He points out there are more women than men in say Education while the reverse is true in STEM fields; and, the more egalitarian a society appears to be, the greater the difference in gender differences when it comes to career choices; moreover, this is consistent with men preferring “things” and women “people”. A glance at his slides would tell you he did some serious analysis/number crunching using bibliometric data collected from the High Energy search engine INSPIRE. I’m not able to independently verify the chronology of events, but there was outrage on social media regarding his talk; even press coverage; and his talk slides/videos were officially censored and the physicist himself was suspended from CERN itself — see CERN’s press release. (Soon after, Strumia’s funding agency, the European Research Council, as well as his home institute University of Pisa, both initiated an investigation against him.) At the end of his slides, Strumia said

PS: many told me “don’t speak, it’s dangerous”. As a student, I wrote that weak-scale SUSY is not right, and I survived. Hope to see you again.

The closest I could find to a justification of such a drastic action of suspension — I’ve been fired once and nearly fired another time over the course of my own 14-year post-Bachelor’s degree academic career, both times without good reasons, so I know full well how that feels like! — is Strumia’s slide 15, where he compared his citation counts with the female Comissar (as I understood it, who was also the organizer of the conference) and another female physicist whom CERN had recently hired, whereas Strumia himself was not offered the same job. This was apparently construed as “attacks on individuals,” which in turn breached CERN’s Code of Conduct. (If there are any misunderstandings on my part, I’d like to hear it; it’s difficult to decipher precisely what happened using information gleaned from the news media and the outrage-driven social media.) However, Strumia’s slide 15 clearly shows, with links to the INSPIRE database so the reader may readily verify the facts for herself, that he did in fact have an order of magnitude more citations than both women: his 30K versus the women’s 2-3K.

Now, I’m just as sensitive as the next human being, and I do consider such a manner of communication to be a rather blunt one. But the scientific ethos requires that, whenever presented with actual evidence, we should address it head on, and not let our personal offense get the better of us as scientists. Namely, “Why wasn’t Strumia hired when he had ten times more citations than the women?” appears to be a legitimate scientific question here. (On the other hand, I was told there were other hires Strumia omitted, and if so I wish he had put everyone on the list for comparison.) Furthermore, observe that was not even the only point on slide 15: he went on to show, of the CERN fellows present, the males had more citations, research papers and years of experience. To suspend him due to the top half of one slide out of 26; to quickly censor his videos such that concerned members of the scientific community (such as I) and of the public cannot independently ascertain what he expressed verbally; and to coat the press release with platitudes regarding “diversity” — altogether does not bode well for the scientific integrity of the particle physics laboratory on our Planet, when it comes to gender issues. The only physicists I am aware of who have actually tried to re-analyze Strumia (and Torre)’s work is Sabine Hossenfelder and her graduate student Tobias Mistele (though using arXiv data, not INSPIRE ones); I believe that is the only true way to respond constructively to the dialog. The rest, I’m afraid, has merely contributed to the Social Justice Warrior far Left Wing I-am-fuming-mad-and-I-need-no-justification culture that infests much of Western Academia these days. If you have been following the news for the past decade or so, Strumia is only but one of many academics/scientists who have been mobbed due to their non-politically-correct views.

I do not think it is unreasonable to postulate, all subsequent High Energy Theory and Gender workshops at CERN — recall that was the first! — will be saturated with talks that will dutifully tow the “women are oppressed/discriminated against” line. This is what such a harsh treatment of Strumia would produce. As scientists, we really need to do some self-examination and ask: is this the scientific outcome we wish to see if truth and intellectual integrity are to prevail? I’m sure it is possible to find sexist individuals, but if calling into question the mainstream narrative of systemic discrimination against women in STEM disciplines is considered taboo, then we have lost our way as scientists.

To be able to think critically through any important issue — particularly complicated and sensitive ones such as gender differences — it is paramount that one is able to hear from and debate against a broad range of views. This way, their relative strengths and weaknesses may be weighed and rational responsible thinkers could propose ideas based on the best available information at hand. This is why freedom of speech is fundamental to any serious democracy. Specifically, it is precisely to allow for contrarian views — popular ones don’t fear backlash from public and/or government persecution! — that is why liberal Western democracies, of which the US is a prime example, provides legal protection for the freedom of expression. (The US Constitution has enshrined this right within its First Amendment.) However, this freedom of expression should not be mere government law. Every one of us is responsible for upholding the right atmosphere within the organizations/societies we belong to, if we wish for there to be an uninhibited exchange of ideas, in order to approach the truth as closely as possible. I want to put on the record, this was why I was motivated to sign the following petition I found online:

CERN: Return Prof. Strumia to office!

Petition to Fabiola Gianotti, Director General CERN, Geneva

Professor Alessandro Strumia, CERN, spoke on Friday 28th September 2018 at a workshop in Geneva on gender and high energy physics. In his presentation he provided evidence for employment policies in physics that were discriminatory toward men and data supporting his opinion that women were given advantages in the academic world purely on the basis of their gender.

As a result, Professor Strumia was suspended with immediate effect by CERN on the grounds that his remarks were antithetical to its code of conduct and to its values.

There can be no free research and freedom of expression if any person must live in fear of existential threat simply for expressing his or her opinion.

Quite independently of the truth of Professor Sturmia‘s statements, none of them can be construed as defamatory, insulting or discriminatory. The opinion he expresses has been expressed many times in multiple research papers and by many other men and women of professional standing.

We cannot and will not tolerate opinions being censored simply because they are in contradiction with mainstream opinion. To do so would be to encourage a totalitarian trend our democracy should not allow.

As an example of political bias in Academia, the particlesforjustice letter — which even contains a thinly veiled threat to destroy Strumia — was posted on the Facebook group Astronomers, whose members are primarily professional astronomers / astrophysicists / physicists. This passed moderation despite the explicit rule that political and non-scientific postings are prohibited. While commenting against the letter, I was challenged to set up my own petition. Even though I did not do so, I did find the above petition and decided to post it in response — the commentary was later shut down by one of the moderators simply because the petition “did not originate in the scientific community and is not appropriate for this forum”. My private messages to the moderators have thus far not been replied to. Ironically, soon after that, someone posted a link to the selection committee for the Breakthrough and New Horizons Prize in Fundamental Physics — and, instead of celebrating the breadth and depth of the scientific expertise assembled — outrage ensued regarding the lack of women on the panel. Of course, no moderation whatsoever was imposed, despite the highly political and un-scientific nature of the discussion.

Update 19 November 2018: I found a very careful article written by a high energy physicist debunking many of the points raised by the particlesforjustice “Community” letter I linked to above. It speaks to the sad state of affairs in the Physics and Astrophysics communities that the author felt the need to remain anonymous.

Update 4 December 2018: There is now another article rebutting the particlesforjustice letter.

Yet another “gender bias” article (this time from Nature) was posted on the Facebook group Astronomers, and I tried to challenge the mainstream gender ideology narrative. This got me banned from the group permanently. An old classmate of mine from graduate school wrote to me to tell me she has unfriended me on Facebook because my comments were “problematic”.

Update 8 March 2019: Strumia has been ousted from CERN; see the updated press release here. (Strumia’s home institute, the University of Pisa, has also issued a public sanction against him here.) I wonder how many people (and their families) funded by Strumia’s ERC grant are going to be affected by this action?

Update 15 March 2019: From Strumia himself — see here.

Update 14 April 2019: Someone has now put together the audio and the slides of Strumia’s talk here.

Update 20 April 2019: See here and here.


Is Strumia a crackpot when it comes to the science of gender discrimination in STEM fields? Is there the slightest possibility that men could be discriminated against in STEM disciplines such as High Energy Theory? (Update: In a subsequent interview Strumia had on the Saad Truth youtube channel, he said he did not think men were discriminated against. IMHO, I think he was being generous — if further studies corroborate his findings that women are shown preference in faculty-level hiring, does it not stand to reason that men are therefore discriminated against, for a fixed number of available jobs?) That his science is horrendous has been asserted repeatedly in the above letter and throughout social media. I cannot speak to the detailed analysis he had done; but I have been aware, since a few years ago and also cited by Strumia in his slides, that Williams and Ceci (faculty at the Department of Human Development, Cornell University) had found a preference for hiring women over men at the tenure-track level. In their youtube video, they also debunked the mainstream claim there is a ton of evidence to support discrimination against women — at least when it came to hiring them as professors.

“… We were really quite shocked, in poring over this literature — it took us many months to digest it all — how little evidence there was. And in fact, there was no experimental evidence. There were experiments, many of them, showing sex biases in hiring, but not of professors, not of tenure-track professors. … [After Wendy M. Williams spoke.] … But there actually was a lot of actuarial evidence that actually went opposite to the bias claims. By that I mean, there were a lot of very large scale studies that looked at who got hired. And these studies — again going back to the mid 1980’s — showed that, over and over again, women were hired at a higher rate than their fraction of the applicant pool. So women were less likely to apply for jobs in math intensive areas, but if they did apply they were more likely to be interviewed and more likely to be hired. … ” — Stephen J. Ceci

One thing I wish they had done was to include physics and/or astrophysics in their analysis. (Of all the “math intensive” groups they analyzed, only male economists were gender neutral.)

Update: See also here.


My own sense is that my fellow scientists really need to take a hard look at the planks in their own eyes and recognize they are — whether consciously or not — taking part in science denial themselves, despite often ridiculing the Right for climate-science denial. Evolutionary forces have shaped how the genders metamorphosed throughout humanity’s existence, due to the different roles they have played for the majority of that duration; and, hence, it would be shocking if men and women were truly identical. I urge the open minded amongst my fellow scientists: please, educate yourselves a tad. (That includes myself, of course — I am no expert in evolutionary biology.) In particular, women and men on average have different interests, life priorities; and therefore make distinct career choices. There really is no good reason to expect a 1:1 ratio in women to men in various careers, such as Physics versus Nursing, and to force it so would in fact necessarily involve discrimination.*

Fortunately, for the busy/lazy physicist / astrophysicist / astronomer out there, there are now plenty of readily accessible youtube videos discussing gender differences known to science, from the experts themselves. (If readers wish to contribute more links, please do post them in the comments section below.)

Let’s begin with the cognitive psychologist Steven Pinker, who wrote the book The Blank Slate: The Modern Denial of Human Nature. Here, Pinker tells us men tend to “chase status at the expense of family” whereas women tend to value family over career. Women gravitate towards “people-oriented” careers whereas men towards “things-oriented” ones — even at the PhD level, more women are pursuing degrees in Education than in Physics, say; even though the total number women pursuing higher education has been growing significantly over the past decades.** Men tend to be the risk-taking ones. Men are better at three dimensional mental rotations, spatial perception and visualization. “Women are better at mathematical calculation” and “men score better on mathematical word problems and tests of mathematical reasoning”. Pinker goes on to explain why there are good reasons to believe many of these sex differences are biological; i.e., they cannot be accounted for solely due to “socialization”. There are large differences in exposure to sex hormones starting prenatally; and small differences in size, density, cortical asymmetry, hypothalamic nuclei of men versus women’s brains. Gender differences in personality transcends “ages, years of data collection, education levels, and nations”. Many of these gender differences has not changed with time; are also seen in other animals; and in fact emerge in early childhood. He even spoke about cases where boys without penises (due to accident or otherwise) and who were brought up as girls, still ended up exhibiting male typical behavior. He also advertises other popular level books like his, that explains the scientific evidence for the biological factors behind gender differences. Steven Pinker can also be found speaking with Dave Rubin here and here about related issues.

Debra Soh, who has a PhD in neuroscience, has been discussing how far Left politics has made discussing the science of transgendered people very difficult, even within academia. Herehere, and here (among other similarly humor-tinged clips) she explains that exposure to testosterone before birth (i.e., “prenatal exposure”) have serious impacts on why the different genders develop different interests. Higher levels lead to “male-typical activities” such as mechanical stuff; whereas lower levels are associated with “socially-engaging” ones. Women are higher in agreeableness and neuroticism, and lower in stress tolerance. (“Neuroticism is simply a technical term for someone’s likelihood to experience negative moods,” according to Soh.) Rates of depression are higher in women. Testosterone is related to greater risk-taking by men. When it comes to brain structure, certain portions are larger in men than in women; there are more front-to-back connections in men’s brains but more left-to-right-hemisphere connections in women’s brains. She goes on to femsplain why James Damore (who was fired by Google for his now infamous memo regarding his reading of what the scientific literature says about gender differences) in fact got his facts/scientific literature right.

Gad Saad, who founded the field of evolutionary psychology applied to marketing and consumer behavior, runs a youtube channel to counter what he likes to call the “tsunami of lunacy crashing against the shores of reason” — i.e., politically correct culture that has become so illiberal and irrational — can be found speaking about gender differences, for instance, here, here, and here.

Heterodox academy, which was founded by NYU psychologist Jonathan Haidt and others, in an effort to counter the strong left wing illiberal culture of Western academia, contains a page on the abovementioned “Google memo”. They performed a literature review to examine how robust Damore’s claims were. Towards the end of this page,

In conclusion, based on the meta-analyses we reviewed and the research on the Greater Male Variability Hypothesis, Damore is correct that there are “population level differences in distributions” of traits that are likely to be relevant for understanding gender gaps at Google and other tech firms. The differences are much larger and more consistent for traits related to interest and enjoyment, rather than ability. This distinction between interest and ability is important because it may address  one of the main fears raised by Damore’s critics: that the memo itself will cause Google employees to assume that women are less qualified, or less “suited” for tech jobs, and will therefore lead to more bias against women in tech jobs. But the empirical evidence we have reviewed should have the opposite effect. Population differences in interest and population differences in variability of abilities may help explain why there are fewer women in the applicant pool, but the women who choose to enter the pool are just as capable as the larger number of men in the pool. This conclusion does not deny that various forms of bias, harassment, and discouragement exist and may contribute to outcome disparities, nor does it imply that the differences in interest are biologically fixed and cannot be changed in future generations.

If our three conclusions are correct then Damore was drawing attention to empirical findings that seem to have been previously unknown or ignored at Google, and which might be helpful to the company as it tries to improve its diversity policies and outcomes.

There was also a Quillette article written by 4 scientists —  Jussim, Schmitt, Miller, and Soh — on James Damore’s “Google Memo”.

Ellis et al.: there is an entire book — Sex Differences: Summarizing More than a Century of Scientific Research.


Let me close with the following two examples which I find illustrative of the current far-Left Wing culture within the West and its Academy.

Physics postdoc Jess Wade’s Twitter post and her New Scientist article have both compared Strumia’s talk to the memo Damore put out, as if that would decisively rule out any credibility in Strumia’s presentation. For instance, in her New Scientist article, she states:

Unlike my talk, backed by evidence, he [Strumia] cited a bunch of poorly thought out gender science from right-wing thinkers. These included James Damore, who was fired from Google last year for holding similar views.

Is using bibliometric data from INSPIRE slide after slide the same as citing “a bunch of poorly thought out gender science from right-wing thinkers”? More importantly, why did she grossly misrepresent Strumia’s talk as “un-backed by evidence”? In her Twitter thread, Jess Wade was challenged on whether she had read Damore’s memo, but as of this writing I could not find any substantive response from her end. I’m sorry to state, it is quite clear who is being ideologically driven — and it is not Strumia nor Damore, as far as I can tell. And, given the current climate, it is also highly unlikely for her to face any serious push back from other physicists. (Update: Wade appeared in Nature’s 10 as a “Diversity Champion”. In the article she openly misrepresented Strumia’s position, asserting that he was “telling a room of mainly young woman scientists that they’d only ever achieve success in physics due to affirmative action”. Nature then highlighted her criticizing him on Twitter, and in the same breath went on to state Strumia “has been suspended from his work with CERN while an investigation is ongoing”.)

Another example comes from no less than the former governor of Vermont, Howard Dean. At an event at Kenyon College, Dean misrepresented not only the awful conduct of Yale students towards faculty Nicholas and Erika Christakis; but also what James Damore said about women in STEM careers. I was glad to see both Steven Pinker and Heather Mac Donald pushing back with the relevant facts; but as far as I could tell, Dean was simply unwilling to concede his serious errors.

The lack of intellectual integrity and honesty exhibited by the Left when it comes to gender issues is precisely the evidence for its commitment to ideology. From the outrage mob that quickly followed the news of Strumia’s talk, it is clear the strongly illiberal tendencies of the far Left has infiltrated Physics / Astrophysics.

The illiberal, irrational, gender-science-denying and identity politics obsessed character of the Left Wing, which Western Academia firmly belongs, form the key impetus behind why I no longer wish to consider myself part of it.


* I believe the Scientific Method requires that, if we are interested in attracting the most competent and creative scientific minds, people should be judged based solely on merit — for e.g., their past accomplishments. I’m afraid I do find a lot of these academic “diversity” initiatives to be primarily identity politics driven, and not merit driven. If we are genuinely keen in “diversity” we should be investing in thought diversity.

** Overall, in the West, women now outnumber men when it comes to obtaining advanced degrees. See here, for example.

Statement of Author Contributions

What are the actual scientific values practiced by the (theoretical) physics community? In particular, how seriously do we think there should be proper accounting of the intellectual effort of each scientist involved — regardless of seniority — in a given project? Namely, who actually did the damned work? In addition to who came up with the ideas, who wrote the code/did the calculations that made those ideas reality? Who found the means to overcome the technical difficulties? Who carried out the check(s) of the results? Does one’s supervisor deserve to be on one’s paper just because the supervisor is paying the salary; i.e., can intellectual credit be bought? What are the incentive structures in place that reflect our values?

Physical Review D (PRD) — supposedly one of the most highly regarded journals in high energy theory, cosmology and gravitation — recently sent out a survey regarding its “performance”: see here if you’re interested in participating. Now, one of the primary purposes of journals is that of conducting peer review, but I have personally found the process itself to be quite arbitrary and frankly mostly a going-through-the-motion just to acquire a superficial “stamp of approval”. For papers from mid-1990’s onward, I go to the arXiv, not the journals, to read them. If the sub-standard level of refereeing in theoretical physics I have experienced is reflective of the current climate, and if it continues not to improve substantially, I do fear journals will become obsolete in the near future.

One feedback I gave PRD concerned that of intellectual credit. Paraphrasing myself, I wish to advocate:

That theoretical physics journals mandate a description of contributions from each and every individual author on the paper.

I believe both Science and Nature require it — so, I have never understood, why not the top theoretical physics journals? To play devil’s advocate, I am curious: what potential negative consequences could such a mandate have on research collaborations?

What happens to the simple harmonic oscillator path integral at half periods?

Previously we had stated but not elucidated the result for the transition amplitude of the simple harmonic oscillator (SHO) at half periods. The SHO Hamiltonian, with X and p denoting the position and momentum operators respectively, reads

(1):    H_\text{SHO} \equiv \frac{1}{2} p^2 + \frac{1}{2} \omega^2 X^2 .

In most quantum mechanics textbooks, we are told that the transition amplitude from the spatial location x' to x over the time period t-t' is

(1′):    \langle x \vert  \exp[-i(t-t') H_{\text{SHO}}] \vert x' \rangle = \sqrt{ \frac{\omega}{2\pi i \sin[\omega(t-t')]} }  \exp\left[ \frac{i\omega}{2\sin[\omega(t-t')]} \left((x^2+x'^2)\cos[\omega(t-t')]-2xx'\right) \right] .

One can see that, when t-t' equals some multiple of \pi/\omega, this formula does not make much sense.

The classical solution for X is a linear combination of \sin[\omega t] and \cos[\omega t]. The period T is therefore \omega T = 2\pi \Rightarrow T = 2\pi/\omega. Over a half period \pi/\omega, i.e., upon replacing t \to t \pm \pi/\omega, we have \sin[\omega t] \to - \sin[\omega t] and \cos[\omega t] \to - \cos[\omega t]. In other words, starting at time t the motion performs a parity flip X \to -X over a half-oscillation.

We will now discuss why the quantum SHO particle does exactly the same thing. If it begins from the location x' at time t, then it has to be at x \equiv (-)^n x' at time t + n \pi/\omega for integer n. Namely, we have the path integral (aka transition amplitude)

(2):    \langle x \vert \exp[-i H_{\text{SHO}} \Delta t_n ] \vert x' \rangle = \frac{1}{i^n} \delta[x - (-)^n x'] ;

with the time interval

(2′):     \Delta t_n \equiv \frac{n\pi}{\omega} .

One might otherwise think there is an inherent ‘fuzziness’ to quantum motion, but at half-periods, all the quantum aspects of the dynamics seem to make their appearance only in the Maslov index 1/i^n multiplying the Dirac delta-function on the right hand side of eq. (2). As we will now see, eq. (2) is a direct consequence of the exact invariance of the SHO Hamiltonian in eq. (1) under parity. If P is the parity operator that implements P X P^{-1} = -X, then the commutator [P, H_{\text{SHO}}] vanishes.  Therefore the SHO energy eigenstates must be simultaneous eigenstates of the parity operator; specifically, the \{ \vert E_\ell \rangle \} in

(2′):     H_{\text{SHO}} \vert E_\ell \rangle = \left(\ell + \frac{1}{2}\right) \omega \vert E_\ell \rangle

obey

(3):     P \vert E_\ell \rangle = (-)^\ell \vert E_\ell \rangle, \qquad \ell = 0,1,2,\dots.

Now, by inserting a complete set of energy eigenstates, we may begin from the left hand side of eq. (2), to consider motion over a single half period \Delta t_1 = \pi/\omega:

(4):    \langle x \vert \exp[-i H_{\text{SHO}} (\pi/\omega)] \vert x' \rangle = \sum_{\ell=0}^\infty \langle x \vert E_\ell \rangle \langle E_\ell \vert x' \rangle \exp[-i (\ell+1/2) \pi] .

Observe that e^{-i\pi/2} = 1/i; whereas e^{-i\ell \pi} = (-)^\ell. Invoking the parity property of the energy eigenstates in eq. (3) to say e^{-i\ell \pi} \langle x \vert E_\ell \rangle = \langle -x \vert E_\ell \rangle, we may thus write

(4′):    \langle x \vert \exp[-i H_{\text{SHO}} (\pi/\omega)] \vert x' \rangle = \frac{1}{i} \sum_{\ell=0}^\infty \langle -x \vert E_\ell \rangle \langle E_\ell \vert x' \rangle .

The completeness relation in the position representation is \sum_\ell \langle u \vert E_\ell \rangle \langle E_\ell \vert v \rangle = \delta[u-v]; comparing this to eq. (4′) we arrive at

(5):    \langle x \vert \exp[-i H_{\text{SHO}} (\pi/\omega)] \vert x' \rangle = \frac{1}{i} \delta[(-) x - x'] = \frac{1}{i} \delta[x - (-) x'] .

By recognizing the unitary nature of the time-evolution operator \exp[-i H_{\text{SHO}} t ], and applying the result in eq. (5) n times to evolve a quantum particle from x' to some other location x over n half periods, one will recover the primary result in eq. (2).


Reference

  • P. A. Horvathy, “The Maslov correction in the semiclassical Feynman integral,” Central Eur. J. Phys. 9, 1 (2011) doi:10.2478/s11534-010-0055-3 [quant-ph/0702236].