Total Solar Eclipse, 21 August 2017

At roughly 10:25 am August 21 2017 local time, the umbra of the Moon’s shadow swept over Huntington, Oregon, USA. I was fortunate to be there to enjoy this absolutely spectacular event:

SolarEclipse2017_2017_08_21_0036

 

Technical details for the (astro)photography enthusiast: ISO 100, 1/250 s @ f/8; shot with the Canon EF 400mm f/2.8L IS II USM Lens coupled to the Canon Extender EF 1.4x III, mounted on the Canon 7D Mark II camera body. This shot was taken at the end of totality.

Fine tuning          Total Solar Eclipses point to a percent-level “fine-tuning” problem theoretical physicists need to address urgently! 😉 Lifting the numbers from the Wikipedia article here, let us note that the Sun’s apparent angular diameter in the sky ranges from 31′ 31″ to 32′ 33″; whereas that of our Moon runs from 29′ 20 ” to 34′ 6″. (Here ‘ \equiv arcminute(s) and ” \equiv arcsecond(s).) How could the Sun and Moon possibly have such similar apparent sizes as viewed from the Earth’s surface?

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Peierls’ Brackets, Green’s Functions, and Equal-Time Commutators

1. Motivation          In non-relativistic quantum mechanics, there exists in Cartesian coordinates the famous commutation relation between the position X^i and momentum operator P_i, namely,

\left[ X^i, P_j \right] \equiv X^i P_j - P_j X^i = i \delta^i_{\phantom{i}j}.

What is the corresponding commutator in relativistic — i.e., Minkowski-spacetime — quantum field theory (QFT)? For a scalar field \varphi, say, we often state, in terms of Cartesian coordinates that parametrize an inertial frame,

(1.1):     \left. \left[ \varphi[t,\vec{x}], \frac{\partial \mathcal{L}}{\partial (\partial_{0} \varphi)}[t',\vec{x}'] \right] \right\vert_{t=t'} = i \delta^{(d-1)}\left[ \vec{x}-\vec{x}' \right],

where \mathcal{L} is the Lagrangian density and hence \partial\mathcal{L}/\partial (\partial_{0} \varphi) is the conjugate momentum to \varphi.

However, equation (1.1) — with its explicit dependence on the time-derivative of \varphi — makes opaque the underlying Lorentz symmetry enjoyed by the field theory itself: namely, why isn’t there an “equal” treatment of space and time? This issue becomes more pressing when quantizing field theories in curved spacetimes, where the need for a generally covariant description of the field theory does not sit well with the need, in “canonical quantization”, for imposing a quantization condition at “equal times”. I believe it was Peierls who first advocated, through the brackets now named after him, a different route to quantization that allows the process to begin from a generally covariant stand-point. Peierls’ brackets have since been promoted repeatedly by Bryce DeWitt — see, for instance, his two-volume text on Quantum Field Theory. Here, I will follow the discussion in Rafael Sorkin’s lecture notes, to quantize a linear scalar field in curved spacetimes using the difference between its generally covariant classical retarded and advanced Green’s functions. In spacetimes where one can find a global timelike vector field u^\mu, which we shall normalize to unity, i.e., u^2 \equiv u_\mu u^\mu = 1, I will then show how to recover the conventional “equal-time” commutation relations.


2. Peierls’ bracket for a linear scalar field          Let us begin with a scalar field \varphi subject to the following (Heisenberg-picture) equations-of-motion:

(2.1):     \left( \Box + \xi \mathcal{R} \right) \varphi = 0,

where \Box \varphi = \partial_\mu \left( \sqrt{|g|} g^{\mu\nu} \partial_\nu \varphi \right)/\sqrt{|g|} is the wave operator; g_{\mu\nu} is the spacetime metric; |g| is the absolute value of its determinant; \xi is some arbitrary real constant; and \mathcal{R} is the Ricci scalar of the same geometry described by g_{\mu\nu}.

To quantize this theory in a curved spacetime we shall postulate that the field operator \varphi obeys the generally covariant commutation relation

(2.2):      \Big[ \varphi[x], \varphi[x'] \Big] = -i \Delta[x,x'],

where x and x' are now shorthand for spacetime coordinates; and \Delta[x,x'] is defined as the difference between the retarded G^\text{(ret)} and advanced G^\text{(adv)} Green’s functions:

(2.3):      \Delta[x,x'] \equiv G^\text{(ret)}[x,x'] - G^\text{(adv)}[x,x'],

with (cf. eq. (2.1))

(2.4):      \left( \Box + \xi \mathcal{R} \right) G^\text{(ret)} = \left( \Box + \xi \mathcal{R} \right) G^\text{(adv)} = \frac{\delta^{(d)}\left[x-x'\right]}{\sqrt[4]{g[x] g[x']}}.

The left hand side of eq. (2.4) holds with respect to both x and x'. Moreover, the retarded Green’s function G^\text{(ret)}[x,x'] is characterized by the condition that, for a fixed source spacetime position x', it is non-zero only when the observer spacetime location x lies to the future of all appropriately defined constant-time surfaces containing x'. Similarly, the advanced Greens function G^\text{(adv)}[x,x'] is characterized by the condition that, for a fixed x', it is non-zero only when x lies to the past of all appropriately defined constant-time surfaces containing x'.

Anti-symmetry          Because G^{\text{(ret)}}[x',x]=G^{\text{(adv})}[x,x'], we may see from eq. (2.3):

(2.5):      \Delta[x,x'] = -\Delta[x',x].


3. Klein-Gordon product          When dealing with QFT in some d-dimensional curved spacetime, one would meet the indefinite “inner product” of Klein-Gordon (KG). For a pair of scalar functions f and g, their KG product on some constant-time hypersurface \Sigma[t], which we also assume is orthogonal to some unit norm future-directed timelike vector u^\mu, is defined as:

(3.1):      \left( f \odot g \right)[t] \equiv \int_{\Sigma[t]} d^D \Sigma_\mu \left( (\nabla^\mu f) g - f (\nabla^\mu g) \right).

Here, if one may endow the \Sigma[t] surface with D \equiv d-1 spatial coordinates \vec{y} such that the induced metric on the former is h_{ij}[\vec{y}], the directed volume/area element is then

d^D \Sigma_\mu \equiv d^D\vec{y} \sqrt{\det h_{ij}[\vec{y}]} u_\mu[t,\vec{y}],

where u_\mu[t,\vec{y}] means the timelike vector u_\mu is evaluated at the time t, corresponding to the constant-time hypersurface \Sigma[t], and the spatial position \vec{y}.


4. A lemma          To recover equal-time commutation relations familiar from canonical quantization procedures, we first need the following lemma involving the KG product of \Delta in eq. (2.3) and some arbitrary scalar function f.

Let \Delta[x,x'] and f[x'] vanish at the (spatial) boundaries of \Sigma[t] for all relevant times t; i.e.,

\Delta\left[ x,x' \in \partial \Sigma[t] \right] = f\left[ x' \in \partial \Sigma[t] \right] = 0, \qquad \forall t.

Furthermore, let f obey the classical Klein-Gordon equation (cf. eq. (2.1)),

(4.1):     \left( \Box + \xi \mathcal{R} \right) f = 0.

Then,

(4.2):     \left( \Delta \odot f\right)_x[t] \equiv \int_{\Sigma[t]} d^D \Sigma_{\mu'} \left( \nabla^{\mu'} \Delta[x,x'] f[x'] - \Delta[x,x'] \nabla^{\mu'} f[x'] \right) = -f[x].

(The prime on the spacetime indices indicates the associated spacetime coordinate is x'; for e.g., \nabla^{\mu'} \equiv \nabla^{x'^\mu}.)

Proof          To see this, we first suppose x lies to the future of \Sigma[t]. Then, only the retarded portion G^{\text{(ret)}} of \Delta in eq. (2.3) contributes to the integral, namely

\left( \Delta \odot f \right)_x[t] = \int_{\Sigma[t]} d^D \Sigma_{\mu'} \left( \nabla^{\mu'} G^{\text{(ret)}}[x,x'] f[x'] - G^{\text{(ret)}}[x,x'] \nabla^{\mu'} f[x'] \right).

We may convert this integral into a “closed-surface” one by introducing another constant-time hypersurface \Sigma[t_>] that lies to the future of x. For, since the retarded Green’s function G^{\text{(ret)}}[x,x'] = 0 whenever x' lies on \Sigma[t_>],

\left( \Delta \odot f\right)_x[t] = \left(\int_{\Sigma[t]} - \int_{\Sigma[t_>]}\right) d^D \Sigma_{\mu'} \left( \nabla^{\mu'} G^{\text{(ret)}}[x,x'] f[x'] - G^{\text{(ret)}}[x,x'] \nabla^{\mu'} f[x']\right).

That this is now a “closed-surface” integral is because we have assumed \Delta[x,x'] (viewed as a function of x') and f[x'] both vanish on the spatial boundaries of \Sigma[t] for all times t. Since, by assumption, u^\mu is “pointing” towards the future, we have Gauss’ theorem informing us that

\left( \Delta \odot f\right)_x[t] = - \int_{t}^{t_>} d^d x' \sqrt{|g[x']|} \nabla_{\mu'} \left( \nabla^{\mu'} G^{\text{(ret)}}[x,x'] f[x'] - G^{\text{(ret)}}[x,x'] \nabla^{\mu'} f[x'] \right),

where the limits t and t_> remind us the domain of the spacetime-volume integral \int d^d x' \sqrt{|g[x']|} is bounded by the constant-time hypersurfaces \Sigma[t] and \Sigma[t_>]. At this point, by expanding out the covariant derivatives, one should find

\left( \Delta \odot f\right)_x[t] = - \int_{t}^{t_>} d^d x' \sqrt{|g[x']|} \left( \Box_{x'} G^{\text{(ret)}}[x,x'] f[x'] - G^{\text{(ret)}}[x,x'] \Box_{x'} f[x'] \right).

By adding and subtracting \xi \mathcal{R}[x'] G^{\text{(ret)}}[x,x'] f[x'] within the integrand, followed by using the equations-of-motion equations (2.4) and (4.1), one would find

\left( \Delta \odot f\right)_x[t] = - \int_{t}^{t_>} d^d x' \sqrt{|g[x']|} \frac{\delta^{(d)}[x-x']}{\sqrt[4]{g[x]g[x']}} f[x'].

Since x always lies within the integration domain \int_{t}^{t_>} d^d x' \sqrt{|g[x']|} we have arrived at eq. (4.2).

For the case where x lies to the past of \Sigma[t], only the advanced Green’s function now contributes to the integral (\Delta \odot f)_x[t]. A similar line of reasoning — introducing a hypersurface \Sigma[t_<] lying to the past of x — can be used to convert (\Delta \odot f)_x[t] into a “closed-surface” integral, from which eq. (4.2) would again be established once equations (2.4) and (4.1) are employed.


5. Equal-time commutators          Let the above constant-time hypersurface \Sigma[t] be parametrized by D \equiv d-1 spatial coordinates and let \vec{y} and \vec{y}' denote two distinct locations on \Sigma[t]; also denote the “time-derivative” of \varphi as

\dot{\varphi} \equiv u^\mu \nabla_\mu \varphi.

The following equal-time commutation results hold on the equal-time hypersurface \Sigma[t]:

(I) The curved spacetime QFT analog of the quantum mechanical [X^i, P_j] = i\delta^i_j reads

(5.1):     \Big[ \varphi[t,\vec{y}], \dot{\varphi}[t,\vec{y}'] \Big] = i \frac{ \delta^{(D)}[\vec{y}-\vec{y}'] }{\sqrt[4]{\det h_{ab}[\vec{y}] \det h_{a'b'}[\vec{y}']}} ;

(II) while the “equal-time” auto-commutator of the fields and their time derivatives are zero, namely

(5.2):     \Big[ \varphi[t,\vec{y}], \varphi[t,\vec{y}'] \Big] = \Big[ \dot{\varphi}[t,\vec{y}], \dot{\varphi}[t,\vec{y}'] \Big] = 0.

Proof          Let f and g satisfy the KG equation (4.1), but are otherwise arbitrary. (Since they obey second-order-in-time differential equations, by arbitrary we mean they have arbitrary initial conditions (f,\dot{f}) and (g,\dot{g}) on \Sigma[t].) From the lemma in eq. (4.2), we may note that

\left( f \odot \Delta \odot g\right) = -\left( f \odot g\right).

(One can check that the KG product is associative, in that f \odot \Delta \odot g = (f \odot \Delta) \odot g = f \odot (\Delta \odot g).) This can be written as

(5.3):     \left( f \odot \Delta \odot g\right)[t] \\ = -\int_{\Sigma[t]} d^D\vec{y}' \sqrt{\det h_{a'b'}} d^D\vec{y}'' \sqrt{\det h_{a''b''}} \frac{\delta^{(D)}[\vec{y}'-\vec{y}'']}{\sqrt[4]{\det h_{i'j'}[\vec{y}'] \det h_{i''j''}[\vec{y}'']}} \\ \qquad\qquad \times \left( \dot{f}[x'] g[x''] - f[x'] \dot{g}[x''] \right),

where x' \equiv (t,\vec{y}') and x'' \equiv (t,\vec{y}''). On the other hand, via a direct calculation,

(5.4):     \left( f \odot \Delta \odot g \right)[t] \\= \int_{\Sigma[t]} d^D\vec{y}' \sqrt{\det h_{a'b'}} d^D\vec{y}'' \sqrt{\det h_{a''b''}} \left( \dot{f}[x'] u^{\mu''} \nabla_{\mu''} \Delta[x',x''] - f[x'] u^{\alpha'} u^{\mu''} \nabla_{\alpha'} \nabla_{\mu''} \Delta[x',x''] \right) g[x''] \\ - \int_{\Sigma[t]} d^D\vec{y}' \sqrt{\det h_{a'b'}} d^D\vec{y}'' \sqrt{\det h_{a''b''}} \left( \dot{f}[x'] \Delta[x',x''] - f[x'] u^{\alpha'} \nabla_{\alpha'} \Delta[x',x''] \right) \dot{g}[x''].

Since the initial data (f, \dot{f}) and (g, \dot{g}) are arbitrary, we may now proceed to choose \dot{f} = g = 0, and equate the coefficients of f and \dot{g} in equations (5.3) and (5.4) to deduce

-u^{\mu''} \partial_{\mu''} \Delta[x'=(t,\vec{y}'),x''=(t,\vec{y}'')] =  \frac{\delta^{(D)}[\vec{y}'-\vec{y}'']}{\sqrt[4]{\det h_{i'j'}[\vec{y}'] \det h_{i''j''}[\vec{y}'']}},

where we have further employed the anti-symmetric property of \Delta in eq. (2.5). Multiply both sides of the above equation with i, and utilize Peierls’ bracket in eq. (2.2) to replace -i\Delta[x',x''] with [\varphi[x'],\varphi[x'']]. This brings us to eq. (5.1).

Eq. (5.2), in turn, can be verified by choosing the initial conditions f = g = 0 and \dot{f} = \dot{g} = 0.

A side note: this analysis indicates more general commutation relations can likely be derived by choosing profiles for the initial data (f,\dot{f},g,\dot{g}) more sophisticated than the ones considered here.


6. Example: Minkowski spacetime          As a simple application of this formalism, let us quantize the massless scalar field \varphi in Minkowski spacetime g_{\mu\nu} = \eta_{\mu\nu} \equiv \text{diag}[1,-1,\dots,-1] using Cartesian coordinates x^\mu \equiv (t,\vec{x}) and x'^\mu \equiv (t',\vec{x}').

The Heisenberg equations-of-motion of \varphi is

(6.1):     \partial^2 \varphi \equiv \eta^{\mu\nu} \partial_\mu \partial_\nu \varphi = 0.

Denoting the step function as \Theta, the retarded Green’s function 1/\partial^2 has the following Fourier representation:

G^{\text{(ret)}}[x-x'] = i \Theta[t-t'] \int_{\mathbb{R}^D} \frac{d^D \vec{k}}{(2\pi)^D} \left( e^{-i|\vec{k}|(t-t')} - e^{+i|\vec{k}|(t-t')} \right) \frac{e^{i\vec{k}\cdot(\vec{x}-\vec{x}')}}{2|\vec{k}|}.

The advanced Green’s function 1/\partial^2, on the other hand, has the following Fourier representation:

G^{\text{(adv)}}[x-x'] = -i \Theta[t'-t] \int_{\mathbb{R}^D} \frac{d^D \vec{k}}{(2\pi)^D} \left( e^{-i|\vec{k}|(t-t')} - e^{+i|\vec{k}|(t-t')} \right) \frac{e^{i\vec{k}\cdot(\vec{x}-\vec{x}')}}{2|\vec{k}|}.

Taking their difference thus yields the following version of the Peierls’ quantization condition in eq. (2.2):

(6.2):     \Big[ \varphi[x], \varphi[x'] \Big] = \int_{\mathbb{R}^D} \frac{d^D \vec{k}}{(2\pi)^D} \left( e^{-i|\vec{k}|(t-t')} - e^{+i|\vec{k}|(t-t')} \right) \frac{e^{i\vec{k}\cdot(\vec{x}-\vec{x}')}}{2|\vec{k}|},

where we have used \Theta[t-t'] + \Theta[t'-t] = 1.

The timelike vector field which we may employ to define “time-derivatives” is simply \partial_0 itself,

u^\mu \partial_\mu = \partial_0.

It is now not difficult to check — by direct calculation — that the equal time relations

\Big[ \varphi[t,\vec{x}], \dot{\varphi}[t,\vec{x}'] \Big] = i \delta^{(D)}[\vec{x}-\vec{x}']

and

\Big[ \varphi[t,\vec{x}], \varphi[t,\vec{x}'] \Big] = \Big[ \dot{\varphi}[t,\vec{x}], \dot{\varphi}[t,\vec{x}'] \Big] = 0

are consequences of eq. (6.2), thereby yielding consistency with the general arguments made to derive equations (5.1) and (5.2).

Particle interpretation          From eq. (6.1), we may deduce the general solution

(6.3):     \varphi[t,\vec{x}] = \int_{\mathbb{R}^D} \frac{d^D \vec{k}}{(2\pi)^D \sqrt{2 |\vec{k}|} } \left(  a_{\vec{k}} e^{-i|\vec{k}|t} e^{i\vec{k}\cdot\vec{x}} + a_{\vec{k}}^\dagger e^{+i|\vec{k}|t} e^{-i\vec{k}\cdot\vec{x}} \right),

where a_{\vec{k}} and its adjoint a_{\vec{k}}^\dagger are linear operators. (The 1/\sqrt{2|\vec{k}|} normalization was chosen for technical convenience.)

Plug eq. (6.3) into the left hand side of eq. (6.2), and proceed to equate from both sides of the ensuing equation the coefficients of the basis solutions \{ e^{\pm i |\vec{k}|t} e^{i\vec{k}\cdot\vec{x}} \} and \{ e^{\pm i |\vec{k}'|t'} e^{i\vec{k}'\cdot\vec{x}'} \}. This yields the famous Fourier-space/ladder-operator commutation relations

\left[ a_{\vec{k}}, a_{\vec{k}'}^\dagger \right] = (2\pi)^D \delta^{(D)}[\vec{k}-\vec{k}']

and

\left[ a_{\vec{k}}, a_{\vec{k}'} \right] = \left[ a_{\vec{k}}^\dagger, a_{\vec{k}'}^\dagger \right] = 0.

These are continuum (i.e., \vec{k}-space) analogs of the commutation relations obeyed by the ladder operators in the quantum mechanical simple harmonic oscillator system, and is key to building and interpreting the n-particle states of the QFT.


7. References

  • R.E. Peierls, “The Commutation laws of relativistic field theory,” Proc. Roy. Soc. Lond. A 214, 143 (1952). doi:10.1098/rspa.1952.0158
  • B. S. DeWitt, “The global approach to quantum field theory. Vol. 1, 2,” Int. Ser. Monogr. Phys. 114, 1 (2003).
  • R.D. Sorkin, “From Green Function to Quantum Field,” Int. J. Geom. Meth. Mod. Phys. 14, no. 08, 1740007 (2017). doi:10.1142/S0219887817400072 [arXiv:1703.00610 [gr-qc]].

First post!

Welcome — this is my first foray into blogging! My plan is to post at a rate of once to a few times per month.

I am a theoretical physicist, originally from the island country of Singapore. Beginning Fall 2003, I entered graduate school in the United States, to pursue my research interests in theoretical physics. Getting a doctorate took me through a convoluted route, but I did eventually graduate from Case Western Reserve University Fall 2010. I just finished up a Postdoctoral Associate position at the University of Minnesota Duluth; and I am extremely excited to be moving on to an Associate Professor position at National Central University in Taoyuan, Taiwan. This would also bring me much closer to home.

More than 14 years in (theoretical physics) academia has lead me to develop a strong sense that, while it definitely attracts many bright minds, its reward structure is deeply flawed and the incentives to uphold the highest standards of professionalism, ethics, and intellectual integrity are severely lacking. On a personal level, I have struggled to survive, and the toll on my physical and mental health has been considerable. That hard work, scrupulous conduct, and a good amount of creativity/problem-solving capabilities are rewarded with apparent punishment — this has been a recurring theme throughout my career thus far. Given how academia functions, I suspect I am not alone.

I hope to write about physics proper — the topics will necessarily be as idiosyncratic as the range of theoretical research I have worked on — but I also hope to, on this blog, discuss many of the above-mentioned scientific-ethical issues. Because detailed evidence is extraordinarily important to me, I will be rather explicit about the various (admittedly anecdotal) examples I have witnessed/experienced over the years that have influenced my views. This would likely offend many people — so let’s see how much trouble I’m going to get myself into…

The March for Science movement has recently been triggered by the need to push back against the perceived anti-science socio-political climate of our times. While I very much stand with it in spirit, I would also like to urge my fellow physicists to perform some serious collective self-examination, if we desire to promote an equitable and merit-based way forward for our scientific community. Issues related to scientific integrity/ethics, professional conduct, intellectual credit, and the evaluation of scientists and the value of their work, will — I predict — become evermore acute the more data starved theoretical high energy physics and (certain aspects of) cosmology become; the higher the ratio of physics graduate student intake to postdoc/faculty job openings; and the more intense the ensuing pressure to publish often and to work on fashionable topics. Although I take issue with many of the dogmatic teachings of the world’s religions, within the context here, I believe it is appropriate to express my sentiments by quoting the King James 2000 Bible, Matthew 7:3:

And why behold you the speck that is in your brother’s eye, but consider not the beam that is in your own?

I hope this blog will be a place for personal growth; for me to learn/understand more physics; and, through the comments section, for interacting with thoughtful people.