Struggles In Physics

yizenchu.net

I’ve created a new website by signing up with wordpress.com: yizenchu.net. It is still under construction and will likely be so for the coming year.

I’m grateful for having a completely free web host for the past 2+ decades. Stargazing.net was a host primarily for amateur astronomers, and I used it for both my private website and the amateur astronomy club at Cal. Unfortunately, as of a few weeks ago, it appears to have ceased operations.

For the time being, I’ve uploaded my lecture notes onto github.

Do too few derivatives lead to ill-defined variational principles?

Likely in an upper-level course in classical mechanics, a physicist learns about the calculus of variation for particle mechanics. The basic problem may be stated as follows. Let $L$ (usually dubbed the “Lagrangian”) be a function of the trajectory $\vec{q}[t]$ and its velocity $\dot{\vec{q}}[t]$ ; and suppose the initial position $\vec{q}[t=t_i] = \vec{x}'$ and final position $\vec{q}[t=t_f] = \vec{x}$ are fixed. What is the trajectory $\vec{q}[t_i \leq t \leq t_f]$ that extremizes the action integral

$S \equiv \int_{t_i}^{t_f} L dt$ ?

For instance, what is the shortest distance between two points $\vec{x}'$ and $\vec{x}$ in flat space? In this case $L = \sqrt{\dot{\vec{q}}^2}$ and the answer is of course a straight line.

Generally speaking, extremization means: if $\vec{q}$ is the desired trajectory, any slight perturbation $\delta\vec{q}$ away from it must yield a zero change in $S$ at first order in this perturbation — just as extremizing an ordinary function $f[x]$ means its derivative must be zero at the turning point, $f[x+dx] = f[x] + (1/2) f''[x] dx^2 + \dots$ . Note that, if $\vec{q}$ satisfies the fixed boundary conditions, then the perturbations must vanish at the end points $\delta\vec{q}[t=t_i] = 0 = \delta\vec{q}[t=t_f]$ in order for the total trajectory $\vec{q} + \delta\vec{q}$ to obey the same boundary conditions. Now, a direct calculation would reveal, the first order change of $S$ upon the replacement $\vec{q} \to \vec{q} + \delta\vec{q}$ is

(1) $\delta_1 S = \left[ \delta\vec{q} \cdot \frac{\partial L}{\partial \dot{\vec{q}}} \right]_{t=t_i}^{t=t_f} + \int_{t_i}^{t_f} \delta\vec{q} \cdot \left( \frac{\partial L}{\partial \vec{q}} - \frac{d}{dt} \frac{\partial L}{\partial \dot{\vec{q}}} \right) dt$ ,

where the derivatives of $L$ with respect to position are done so with the velocities held fixed, and vice versa.

We see from eq. (1) that for the first order variation to be zero, $\delta_1 S = 0$ , not only must the “bulk term” involving the integral be zero, so must the boundary terms at $t=t_i$ and $t=t_f$ . The first is what yields the Euler-Lagrange equations

(EL): $\frac{\partial L}{\partial \vec{q}} - \frac{d}{dt} \frac{\partial L}{\partial \dot{\vec{q}}} = 0.$

Unfortunately, once the equations-of-motion are obtained, the boundary terms in eq. (1) are often forgotten. That they vanish is not in fact guaranteed; for, it depends not only on the boundary conditions imposed but the structure of the Lagrangian $L$ itself.

Before moving on, let us note that the equations-of-motion in (1) are second order in time, which implies — taken on their own, and not necessarily arising from an action principle — an explicit solution can be determined by either imposing boundary conditions (as we have done here); or, say, initial values for the position and velocity. Fewer than two conditions or more than two conditions would lead respectively to an under-determined or over-determined problem. Hence, the usual variational principle involving two boundary conditions–one for each end of the trajectory–is consistent with the second-order-in-time character of the Euler-Lagrange equations it yields.

Too many derivatives

The case where the Lagrangian contains too many derivatives appears to be well discussed in the recent literature. If $L$ depends not only on $\vec{q}$ and $\dot{\vec{q}}$ but also $\ddot{\vec{q}}$ and/or higher derivatives, the formula in eq. (1) will need to be appended on the right hand side with additional terms involving derivatives of the Lagrangian with respect to them; more boundary conditions will also be needed. In fact, Ostrogradsky tells us, these higher derivative theories generically lead to runaway (i.e., unstable) solutions.

Within the field theory context, the most famous such example is that of Einstein’s General Relativity. The associated Einstein-Hilbert action is often written as

$S_{\text{EH}} = -(16 \pi G_N)^{-1} \int_{\mathfrak{D}} d^4 x \sqrt{|g[x]|} \mathcal{R}[x] ,$

where $\mathfrak{D}$ denotes the region of spacetime under consideration, $|g|$ is the absolute value of the determinant of the metric tensor, and $\mathcal{R}$ is the Ricci scalar.

The first order variation with respect to the metric, $g_{\mu\nu} \to g_{\mu\nu} + \delta g_{\mu\nu}$ , yields

(EH.1): $\delta_1 S_{\text{EH}} = +(16\pi G_N)^{-1} \int_{\mathfrak{D}} d^4 x \sqrt{|g|} \delta g_{\mu\nu} G^{\mu\nu} \\ + (16\pi G_N)^{-1} \int_{\partial \mathfrak{D}} d^3\Sigma^\mu \left( \nabla^\nu \delta g_{\mu\nu} - g^{\alpha\beta} \nabla_\mu \delta g_{\alpha\beta} \right) ,$

where $G^{\mu\nu}$ is the Einstein tensor and $\int_{\partial \mathfrak{D}} d^3\Sigma^\mu$ denotes the directed 3-volume element of the boundary of the spacetime region under consideration. Notice the boundary terms contain the first derivatives of $\delta g_{\mu\nu}$ , and not of $\delta g_{\mu\nu}$ alone; this can be traced back to the fact that the Ricci scalar itself contains terms with two derivatives acting on the metric tensor. Hence, if we wish to define the variation principle with the metric — and not its derivatives — fixed on the boundary, we need to somehow cancel these terms out. This is precisely the purpose of adding to $S_{\text{EH}}$ the Gibbons-Hawking-York term involving the boundary’s extrinsic curvature.

Too few derivatives?

On the other hand, it appears to me, there is also the much less discussed case of a Lagrangian (density) containing too few derivatives. Consider the following flat spacetime action involving the 4-component Dirac spinor $\psi[x]$ .

(Dirac): $S_D = \int_{t_i}^{t_f} dt \int_{\mathfrak{D}} d^3\vec{x} \left( \frac{i}{2} \psi^\dagger \gamma^0 \gamma^\mu \partial_\mu \psi - \frac{i}{2} \partial_\mu \psi^\dagger \gamma^0 \gamma^\mu \psi - m \psi^\dagger \gamma^0 \psi \right) ,$

where the spacetime volume is the time interval $[t_i,t_f]$ together with the spatial volume $\mathfrak{D}$ ; and the $\{ \gamma^\mu \}$ are the $4 \times 4$ Dirac matrices obeying the algebra $\{ \gamma^\mu, \gamma^\nu \} = 2 \eta^{\mu\nu}$ .

The first order variation of the Dirac action is

(Dirac.1): $\delta_1 S_D = \int_{t_i}^{t_f} dt \int_{\mathfrak{D}} d^3\vec{x} \left\{ \delta \psi^\dagger \gamma^0 \left( i \gamma^\mu \partial_\mu - m \right) \psi + \left( -i \partial_\mu \psi^\dagger \gamma^0 \gamma^\mu - m \psi^\dagger \gamma^0 \right) \delta\psi \right\} \\ + \left[ \frac{i}{2} \int_{\mathfrak{D}} d^3\vec{x} (\psi^\dagger \delta\psi - \delta \psi^\dagger \psi)\right]_{t=t_i}^{t=t_f} + \frac{i}{2} \int_{t_i}^{t_f} dt \int_{\partial\mathfrak{D}} d^3\Sigma_i \left( \psi^\dagger \gamma^0 \gamma^i \delta\psi - \delta\psi^\dagger \gamma^0 \gamma^i \psi \right).$

We may demand that these semi-classical Dirac fields obey Dirichlet boundary conditions on the spatial boundaries. This would set to zero the last group of terms. Furthermore, the “bulk” terms on the first line also imply the Dirac equation

$(i \gamma^\mu \partial_\mu - m) \psi = 0 .$

However, because these are first-order-in-time partial differential equations, it is not possible to simultaneously specify $\psi$ on both the initial $t=t_i$ and final $t=t_f$ time hypersurfaces; once $\psi$ is specified on one, its value is determined on the other. Hence, it is not clear to me that we may demand $\delta \psi = 0 = \delta \psi^\dagger$ for both $t=t_i$ and $t=t_f$ . At the most, when evaluated on the solutions subject to spatial Dirichlet boundary conditions, we have

$\delta_1 S_D = \pm \left[ \frac{i}{2} \int_{\mathfrak{D}} d^3\vec{x} (\psi^\dagger \delta\psi - \delta \psi^\dagger \psi)\right]_{t=t_f (+) \text{ or } t_i (-)} \neq 0.$

Is the variational principle leading to the Dirac (and Weyl) equation well defined?

TensoriaCalc

Tensor calculus is messy business. Not only do tensors, by their very nature, contain multiple components, with each component containing possibly very complicated expressions — just, say, taking their covariant derivatives would further incur numerous terms arising from contraction with the rank-3 Christoffel symbols. Since these calculations are not only long and arduous, whose step-by-step details do not deepen physical insight but are oftentimes merely “mechanical” in character, it has been clear to me (and others) that they ought to be automated with Computer Algebra as far as possible.

Since the 2010’s, I started to develop an easy-to-use Wolfram Mathematica package – which I have named TensoriaCalc (added: GitHub) – to tackle concrete (as opposed to abstract) tensor calculus commonly encountered in General Relativity, cosmology, and field theory. (I have already used it to discover a class of GR-MHD cosmological solutions.) To ensure it is indeed user-friendly, I have taken serious effort to construct a framework that is smoothly consistent with the Wolfram language – for instance, there is no need to declare indices; they are just patterns of the form SuperMinus[_Symbol] (upper indices) or SubMinus[_Symbol] (lower indices) – while being versatile enough to accommodate generic multi-component (aka “indexed”) objects as well as future functionality expansion. Just as Graphics is the Head of all graphics-related objects, at the heart of TensoriaCalc, the Tensor object is the basic atom of all “indexed” or “tensorial” quantities, with all relevant information such as coordinates, indices, component(s), etc. stored as Rule’s within it. Differential geometric operations such as covariant, Lie and exterior derivatives, coordinate transformations, Hodge duals, etc. are first defined by how they act on these Tensor’s.

I was very fortunate to have recruited Wei-Hao Chen during the early part of the 2020’s to work intensively on TensoriaCalc. (I also recruited Vaidehi Varma, who worked with us to carefully document the code and write TensoriaCalc’s manual.) He has helped expand its functionality considerably. For example, CoordinateTransformations in my original version was rudimentary, merely generating a List of transformations of the basis 1-forms and vector fields from a given coordinate transformation. But Wei-Hao’s updated version can now implement coordinate transformations on metrics and generic tensors–including the use of orthonormal basis–as well as ‘induce’ them onto a lower dimensional hypersurface. Highly non-trivial!

Update: The arXiv posting is out!

Download TensoriaCalc

Download TensoriaCalc.nb. Then save it in the desired directory, open the notebook and Save. This creates TensoriaCalc.m. To use TensoriaCalc, load this .m file into your Mathematica session.

I recommend working through TensoriaCalc_UserGuide.nb, where we discuss the 3D Euclidean group; the induced metric on the round 2-sphere and the scalar and vector spherical harmonics as eigenfunctions; 3D vector calculus in Cartesian, polar, and spherical coordinates, and their associated eigenfunctions of the Laplacian; the Poincare lemma applied to the Coulomb electric field and the constant magnetic field; the 4D Poincare group and the scalar wave and Maxwell equations; weakly curved static Newtonian spacetimes; the general (positive, negative and flat spatial curvature) Friedmann–Lemaître–Robertson–Walker cosmology; Birkhoff’s theorem and the Schwarzschild black hole, the associated geodesics, as well as the gravitational redshift and light deflection in the weak field limit; and, very briefly, the rotating Kerr black hole. This hands-on approach should provide a good introduction to TensoriaCalc’s functionality.

Please let me know should you have suggestions, constructive criticisms, bug reports, etc.

Generalized Poisson-Arago Spot and Spacetime Causality

Place an opaque disk on the $(1,2)-$ plane centered at its origin; a point source of light at $(0,0,z')$ for $z'<0$ on one side; and a light-sensitive plate — say, a CMOS detector — parallel to the $(1,2)-$ plane intersecting the $3-$ axis at $(0,0,z)$ for $z>0$ on the other side. In the ancient days, folks debated whether light was comprised of particles or waves. If light were particles, so the argument goes, it would take straight lines — this is the basis of geometric optics — and therefore be obstructed by an opaque disk and the center of the detector at $(0,0,z)$ should not receive any signal at all. (Why can’t the particles bounce off the edges of the disk and reach the center of the detector?) On the other hand, if light were waves, it would be able to spread around the edges of the disk, and some fraction of it would be able to reach the center of the disk.

The Poisson-Arago spot is in fact a prediction of the above “light are waves” argument, based on the Kirchhoff diffraction formula that physics graduate students learn as part of a course on electromagnetism. It states that not only is there a non-zero signal at the center of the detector in the above setup — due to diffraction — there is in fact a bright spot lying in the center of the otherwise darker shadow region due to the obstruction created by the disk itself.

In a recent ~~preprint I posted on the arXiv~~ analysis I carried out with Jia-En Chen — this was his MS thesis work — I turned to examine the Kirchhoff diffraction formula in position spacetime, as opposed to the usual frequency space. This is a warm-up to my attempt to understand gravitational lensing of both electromagnetic and gravitational waves from first principles; to compute their associated Green’s functions and to understand the causal structure of the signals. By causal structure, I mean how these waves propagate both in space and time.

Within Kirchhoff diffraction theory, 3D obstacles are approximated as 2D objects lying on a flat 2D plane — in astrophysics lingo, the “lens screen”. For instance, a basketball would become its cross-section, the above opaque disk, in a manner consistent with the placement of the source and detector. Within this approximation scheme, Jia-En and I asked:

How does the signal emitted from the spacetime point $x' \equiv (t',\vec{x}')$ navigate an arbitrary 2D obstacle or aperture to reach the spacetime point $x \equiv (t,\vec{x})$ on the other side of the lens screen?

The key fact about the propagation of zero-mass signals such as electromagnetic or gravitational waves is that — on a flat 4D spacetime background — they do so strictly on the null cone, where time elapsed is equal to the spatial distance traveled. Therefore, in the absence of obstacles, light rays do indeed travel in straight lines, whether you call them waves or particles. However, in the presence of obstacles, they are able to scatter off the edges of these obstacles and continue along a straight path but in a different direction from before. This scattering picture in position spacetime is what is known as diffraction in frequency space. For a given observer at $x$ and source at $x'$ , we may therefore consider the locus of spacetime points that lie simultaneously on the forward null cone of $x'$ and the backward null cone of $x$ — i.e., demand that causality must hold. This in fact, for a fixed elapsed time $t-t'$ , yields an ellipsoid whose focii are $\vec{x}'$ and $\vec{x}$ . Whenever this ellipsoid is able to intersect the boundaries of this obstacle, there will be a diffracted signal. Whenever there are no intersections, there is no diffraction.

From paper: “*Scattered Signal from $x'$ to $x$ : Spatial Ellipsoid Picture.* The null signals emitted from $x'^\mu = (t',\vec{x}')$ and those received by $x^\mu = (t,\vec{x})$ together defines an ellipsoid … of size $s=t-t'$ in $\vec{x}''-$ space. Since the obstacle — here denoted by the red region and whose boundary is defined by $W=0$ — lies on the 2D lens plane, and since the $z''=0$ planar cross section of the ellipsoid is an ellipse (denoted by the black oval), the scattered signal is non-zero whenever the ellipse is able to intersect the obstacle’s boundary. The left panel shows the early time regime, where $t-t'$ is small enough such that the ellipse lies entirely within the obstacle’s boundary, and therefore no signal is received at $\vec{x}$ . The middle panel illustrates the intermediate time regime, where $t-t'$ has grown such that the ellipse now intersects the obstacle’s boundary. In particular, we have illustrated the discrete case … with 4 distinct intersection points represented by the yellow dots. A null signal emanating from $\vec{x}'$ takes a straight path to hit one of the 4 intersection points, before scattering off to a different straight path to reach $\vec{x}$ . The $G^{+,S}_O$ is the net signal from summing over all four scatterings. On the right panel, depicting the late time regime, $t-t'$ has become so large that the entire ellipse lies outside of the obstacle’s boundary, yielding no signal at $\vec{x}$ .”

Now, these intersections between the ellipsoid determined by causality and the boundaries of the obstacle(s) are usually discrete points; for e.g., four of them, in the above figure. However, if a portion of the boundary forms part or the entirety of an ellipse or circle, then this finite segment — as opposed to the usual discrete points — can be the ellipsoid-boundary intersection. This is, of course, provided the source-observer configuration is adjusted appropriately. When this occurs, we showed that a Dirac $\delta-$ function signal forms at the observer location, which we interpret as the central peak of the Poisson-Arago spot. (Note that we no longer need a perfectly circular disk to form the Poisson-Arago spot, as per the usual setup above.) Discrete points of intersection would instead produce a diffuse signal, not an infinitely peaked one.

Moreover, if we do specialize to the usual setup, where the source is at $(0,0,z'<0)$ , observer at $(0,0,z>0)$ and the disk of unit radius is centered at $(0,0,0)$ — we can work out the signal $G^+_O[x,x']$ explicitly.

The $\delta-$ function argument follows from the Pythagorean theorem — the time elapsed $T \equiv t-t'$ is the amount of spatial distance from the source at $\vec{x}'$ to the edge of the circle plus that from the edge of the circle to the observer at $\vec{x}$ .

Censorship by arXiv

This is my first experience of censorship by the arXiv. I looked up which section of the arXiv the Kirchhoff diffraction formula appears, and not surprisingly, it was physics.optics. Because I have never posted there before, we got a kind physicist to endorse us. After waiting for two whole weeks, we got the following email, devoid of specifics regarding why our paper — discussing a fundamental equation of optics — was not allowed. If we need to get it published in a journal first, why is the arXiv called a “pre-print” server? More importantly: Is this sort of censorship consistent with the scientific ethos?

Update May 2026: I’ve updated the paper and one of the figures above.

Energy-Momentum of Non-Local Gravity Theories

Long range forces such as electromagnetism and gravitation are usually associated with zero mass particles — namely, the photon and the graviton respectively. When quantum effects are included the resulting effective action for these basic force laws become non-local; namely, the Lagrangian density no longer involves the photon or graviton (really, the metric) field at a single point in spacetime, but rather, at more than one location. To preserve the general covariance of the action, this non-locality oftentimes involves the inverse wave operator of the spacetime geometry itself.

Over the course of two papers, in arXiv: 1811.04647 (with Sohyun Park) and 2410.24076 (with Afidah Zurioda), I examined the amount of energy momentum transported by the emission of gravitational waves in the Deser-Woodard I and II, Vardanyan-Akrami-Amendola-Silvestri and Amendola-Burzilla-Nersisyan non-local models of modified gravity theories. These models were inspired either by the need to account for Dark Energy or quantum gravitational effects. What I found was that — provided we take their non-local properties seriously** — the energy-momentum fluxes of gravitational waves at large distances from their material sources do not even make sense: the total energy transported to infinity is, well, infinite.

In more detail, because the area element on the sphere of radius $r$ scales as $r^2$ , we expect the flux to scale as $1/r^2 \equiv$ 1/(source-observer distance $)^2$ , in order for the total flux to be finite at large distances. However, in these models, due to the non-local self interactions of the graviton, the flux instead scales as $1/r, r^0, r^1,$ etc. We extracted the flux in the same way the gravitational wave flux is extracted in General Relativity: Expand the geometric (matter-independent) terms in the equations-of-motion up to 2nd order in the metric perturbations; then, identify the gravitational wave (pseudo-)stress tensor as these second order terms divided by $-8\pi G_N$ . That we obtained a $1/r$ flux in some of these non-local models is not unlike the computation of the first nonlinear corrections to the gravitational perturbation solution in General Relativity itself; for there it necessarily involves a non-local 3-point function, which in turn leads to a $1/r$ contribution.

** Capozziello et al appeared to have avoided these divergent gravitational wave fluxes altogether by simply converting the non-local interactions of the gravitons into local ones, through the introduction of auxiliary field variables — see, for instance, arXiv:2412.13629 and its equation 2.4. However, these auxiliary variables were originally introduced to obtain the final non-local equations-of-motion for these non-local gravity models in a more tractable manner; not as a means to itself, so to speak. In other words, I do not believe these auxiliary variables to be physical ones, if we were to take the non-local properties seriously. Instead, one would likely have to start from the fundamental theory from which these four effective models arose in the first place, to obtain a sensible gravitational wave flux.

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!