nLab quantum commutators are time-ordered ordinary products -- section

Quantum commutators are time-ordered ordinary products of observables

The path integral formulation (and generally the notion of time-ordered products satisfying the Schwinger-Dyson equation) reveals the following foundational fact of quantum physics, which is “well known” but not widely appreciated (most textbooks don’t mention it).

As slogans, in slightly increasing order of accuracy:

Slogan: The quantum (operator) product of observables is their ordinary product after slightly shifting their time domains into operator order.

Or more technically:

Slogan: The operator product $O_2(t) \star O_1(t)$ of observables at equal time $t$ is their ordinary product after slightly shifting the observation $O_2$ to after $O_1$, hence is $\underset{\underset{\epsilon \to_+ 0}{\longrightarrow}}{\lim} O_2(t + \epsilon) O_1(t)$.

Or rather:

Slogan: The non-commutativity of quantum observables (such as witnessed by the canonical commutator between field observables and their canonical momenta) reflects that the temporal order of observation matters, hence reflects the difference $\underset{\underset{\epsilon \to_+ 0}{\longrightarrow}}{\lim} \big( O_2(t + \epsilon) O_1(t) - O_1(t + \epsilon) O_2(t)\big)$.

Here these (limits of) ordinary products of ordinary observables (on $\mathbb{C}$-valued functions of physical configurations) are to be understood as expectation values as produced by a path integral with respect to some (arbitrary) state. We proceed to say this in more technical detail.

This insight goes back to Feynman 1948 p. 381, who considered it in the context of non-relativistic quantum mechanics, reviewed below in:

• In Quantum Mechanics

But this generalizes to relativistic quantum field theory, discussed below in:

• In Quantum Field Theory.

In fact, the analogous statement remains true also in light-front quantization (cf. Rem. below), where it says that the canonical commutators are given by ordinary products of observables after shifting their light-front-parameter domain into operator order.

In Quantum Mechanics

The following is the original observation of Feynman 1948, p. 381.

(This has been recalled by Feynman, Hibbs & Styer 2010 (7.45); Schulman 1981, Ch. 8; Nagaosa 1999, pp. 33; Ong 2012; Rischke 2021 Section 5.6, but all these authors follow Feynman 1948 essentially verbatim. In particular, none actively recognizes the Schwinger-Dyson equation in the argument nor comments on generalization beyond the 1d discretized nonrelativistic path integral that Feynman considered and which we recall now.)

Consider the path integral for a particle propagating on a circle $S^1$, and approximated by an ordinary integral over positions $x_t$ at $N$ discrete time steps $t \in \mathbf{N} \coloneqq \{0, 1, \cdots, N-1\}$, hence over discretized trajectories

$$x \colon \mathbf{N} \longrightarrow S^1 \mathrlap\,.$$

To recall that the quantum expectation value of an observable $O \colon (S^1) ^{\mathbf{N}} \longrightarrow \mathbb{C}$ with respect to a pure quantum state $\psi \colon S^1 \longrightarrow \mathbb{C}$ is expressed as the following (discretized) path integral:

(1)$$\big\langle O \big\rangle \;\coloneqq\; \tfrac{1}{\mathcal{N}} \int O(x) \, \exp\big(\tfrac{\mathrm{i}}{\hbar} S(x)\big) \, \psi^\ast(x_N) \psi(x_0) \, D x \mathrlap{\,,}$$

where

$$\mathcal{N} \coloneqq \int \exp\big(\tfrac{\mathrm{i}}{\hbar} S(x)\big) \, \psi^\ast(x_N) \psi(x_0) \, D x$$

is the normalization factor (the “partition function”), and where

$$\int D x \,\coloneqq\, \int_{S^1} \cdots \int_{S^1} \mathrm{d}x_0 \cdots \mathrm{d}x_{\mathbf{N}-1} \mathrlap{\,.}$$

With that simple setup, ordinary integration by parts gives for an observable which is a partial derivative,

$$O(x) \,=\, \tfrac{\partial F}{\partial x_t} (x) \,, \phantom{--} 1 \lt t \lt N \mathrlap{\,,}$$

that its expectation value is equivalently expressed as:

(2)$$\begin{aligned} \big\langle O \big\rangle & \equiv \big\langle \tfrac{\partial F}{\partial x_t} \big\rangle \\ & -\tfrac{\mathrm{i}}{\hbar} \big\langle F \tfrac{\partial S}{\partial x_t} \big\rangle \end{aligned}$$

(which we may recognize as the 1d discretized form of what is now called the Schwinger-Dyson equation in quantum field theory more generally).

Specializing this to the free non-relativistic particle of mass $m \gt 0$, for which the discretized action functional is

$$S(x) \,=\, \sum_{1 \leq t \lt N} \tfrac{m}{2} ( x_{t+1} - x_{t} )^2 \tfrac{1}{N} \mathrlap{\,,}$$

the key point to observe is that

$$\tfrac{\partial S}{\partial x_t} \,=\, m \tfrac{ (x_{t} - x_{t-1}) }{1/N} - m \tfrac{ (x_{t+1} - t_n) }{1/N} \mathrlap{\,.}$$

Using this when entering equation (2) with the choice

$$F \coloneqq x_t$$

gives:

$$\mathrm{i}\hbar = \big\langle x_t \, m \tfrac{ (x_{t} - x_{t-1}) }{1/N} \big\rangle - \big\langle m \tfrac{ (x_{t+1} - x_{t}) }{1/N} \, x_t \big\rangle \mathrlap{\,.}$$

Here we recognize

$$p_{t+1/2} \coloneqq m \tfrac{ (x_{t+1} - x_{t}) } {1/N}$$

as the discrete approximation to the momentum observable at time $t + 1/2$, in terms of which we have found that:

(3)$$\begin{aligned} \mathrm{i}\hbar & = \big\langle x_t \cdot p_{t - 1/2} \,-\, p_{t + 1/2} \cdot x_t \big\rangle \,. \end{aligned}$$

In the time continuum limit, this becomes

$$\mathrm{i}\hbar \,=\, \big\langle x_t \cdot p_{t - \epsilon} \,-\, p_{t + \epsilon} \cdot x_t \big\rangle$$

for $\epsilon \to 0$.

But this is clearly the path integral expression for what in operator formalism is the canonical commutation relation

$$\mathrm{i}\hbar = \hat x \cdot \hat p - \hat p \cdot \hat x \,.$$

In conclusion, the observable corresponding to a quantum operator product $B \cdot A$ of observables at times $t$ may be thought of as the result of first shifting the temporal supports of the observables so that $B$ is observation at a time just a little after that of $A$, and then forming the ordinary product of observed values.

As Feynman 1948 also noticed, the same conclusion holds with an ordinary potential energy term included in the action functional, since its contribution is non-singular and hence vanishes in the final $\epsilon\to 0$ limit.

In Quantum Field Theory

In fact, by using the Schwinger-Dyson equation, this argument generalizes (cf. physics.SE:685812) from the quantum mechanics of a nonrelativistic particle to general quantum field theories with ordinary potential energy terms, as follows.

(Conversely, the product of observable-values in the path integral corresponds to the time-ordered product of the corresponding linear operators (eg. Polchinski 1998 (A.1.17); Rischke 2021 (5.63).)

Imagine a path integral-formulation exists of some $1+d$-dimensional quantum field theory determined by a Lagrangian density $L$ with an ordinary potential energy term and denote the corresponding expectation values in some state by $\langle-\rangle$ — or else regard $\langle-\rangle$ as denoting the time-ordered product of its arguments, that’s all we need.

Let $\phi$ be one of the field species. (It could be a scalar field but it may just as well be a component of any more complex field.)

Assuming we are on cylindrical Minkowski spacetime $\mathbb{R}^{1,d} / \mathbb{Z}^{d}$ — just for notational simplicity — then the Schwinger-Dyson equation for field insertion $\phi(y)$ says that

(4)$$\bigg\langle \underset { \delta S / \delta \phi(x) }{ \underbrace{ \Big( \frac{ \partial L }{ \partial \phi } (x) - \partial_\mu \frac{ \partial L }{ \partial (\partial_\mu \phi) } \Big) } } \, \phi(y) \bigg\rangle \;=\; \mathrm{i}\hbar \left\langle \frac{ \partial \phi(y) }{ \partial \phi}(x) \right\rangle \;=\; \mathrm{i}\hbar \, \delta^{1+d}(x-y) \mathrlap{\,.}$$

This is the field theoretic version of Feynman’s equation (2) above.

Now consider the integration of this expression in the variable $x$ over the spacetime region in a small time interval $(y^0- \epsilon, y^0 + \epsilon) \times \mathbb{R}^{d}/\mathbb{Z}^{d}$ and let $\epsilon \to 0$. Then:

1. the first summand on the left of (4) vanishes (being asymptotically proportional to $\epsilon$ since we are assuming that the potential term and hence the $\phi$-dependence of $L$ is that of an ordinary smooth function),

2. by Stokes's theorem the spatial integral over the spatial components of the second summand vanishes and

3. the remaining temporal integral of its temporal component gives two boundary terms (where we now decompose $x = (x^0, \vec x)$):

(5)$$\underset{\underset{\epsilon\to 0}{\longrightarrow}}{\lim} \int_{\mathbb{R}^{d}/\mathbb{Z}^{d}} \mathrm{d}^d \vec x \, \left\langle \frac{ \partial L }{ \partial (\partial_0 \phi) } (y^0 + \epsilon, \vec x) \, \phi(y^0, \vec y) \;-\; \frac{ \partial L }{ \partial (\partial_0 \phi) } (y^0 - \epsilon, \vec x) \, \phi(y^0, \vec y) \right\rangle \;=\; -\mathrm{i}\hbar \mathrlap{\,.}$$

Here we recognize the canonical momentum $\pi$ to the field $\phi$:

$$\pi(x) \;\coloneqq\; \frac{ \partial L }{ \partial (\partial_0 \phi) }(x) \mathrlap{\,,}$$

so that

(6)$$\underset{\underset{\epsilon\to 0}{\longrightarrow}}{\lim} \int_{\mathbb{R}^{d}/\mathbb{Z}^{d}} \mathrm{d}^d x \, \Big\langle \pi(y^0 + \epsilon, \vec x) \, \phi(y^0, \vec y) \;-\; \pi(y^0 - \epsilon, \vec x) \, \phi(y^0, \vec y) \Big\rangle \;=\; -\mathrm{i}\hbar \mathrlap{\,.}$$

This is the field-theoretic version of Feynman’s equation (3) above.

We may redo this derivation after multiplication of the original Schwinger-Dyson equation (4) with any “smearing function” $f(\vec x)$ (a spatial bump function). Then where we used Stokes' theorem above we are now faced with an integration by parts that picks up terms proportional to the gradient of $f$ — but if the dependence of $L$ on spatial derivatives of $\phi$ does not have unusual singularities (i.e. if the kinetic energy term in $L$ is a standard one) then these terms vanish with $\epsilon$ just as the potential energy term does, and hence we end up with

(7)$$\underset{\underset{\epsilon\to 0}{\longrightarrow}}{\lim} \int_{\mathbb{R}^{d}/\mathbb{Z}^{d}} \int \mathrm{d}^d \vec x \, f(\vec x) \, \Big\langle \pi(y^0 + \epsilon, \vec x) \, \phi(y^0, \vec y) \;-\; \pi(y^0 - \epsilon, \vec x) \, \phi(y^0, \vec y) \Big\rangle \;=\; -\mathrm{i}\hbar f(\vec y) \mathrlap{\,.}$$

But since this holds for all smearing functions $f$, this is equivalent to the distributional equation

(8)$$\underset{\underset{\epsilon\to 0}{\longrightarrow}}{\lim} \Big\langle \pi(y^0 + \epsilon, \vec x) \, \phi(y^0, \vec y) \;-\; \pi(y^0 - \epsilon, \vec x) \, \phi(y^0, \vec y) \Big\rangle \;=\; -\mathrm{i}\hbar \, \delta^d(\vec x - \vec y) \mathrlap{\,,}$$

which is the claimed incarnation of the canonical commutation relation of field operators at equal times,

$$\big[ \widehat{\pi}(\vec x), \widehat{\phi}(\vec x) \big] \,=\, -\mathrm{i}\hbar \, \delta^d(\vec x - \vec y) \mathrlap{\,,}$$

now re-expressed as an expectation value of ordinary products of observables after shifting their temporal domains into operator order.

Remark

The analogous conclusion holds also for light front quantization, with the role of the time coordinate $x^0$ now played by the light front parameter $x^+$, for

$$x^\pm \coloneqq (x^0 \pm x^d)/\sqrt{2} \,.$$

Here the light-front canonical momentum to a field $\phi$ is (cf. Burkardt 1996 table 2.1 for the following equations):

$$\pi \,=\, \frac { \partial L } { \partial (\partial_+ \phi) } \mathrlap{\,,}$$

which for Lagrangian densities with standard kinetic energy term

$$L \,=\, \partial_+ \phi \, \partial_- \phi \,-\, \tfrac{1}{2} (\vec \nabla_\perp \phi)^2 - V(\phi)$$

comes out as

$$\pi \,=\, \partial_- \phi \mathrlap{\,.}$$

While the nature of this light front momentum in canonical quantization (where it is a second class constraint) is quite different from the nature of the canonical momentum in instant form, at the end the equal-LF-parameter commutation relation has the same form as the usual equal-time commutator:

$$\Big[ \widehat{\pi}\big(x^+, x^-, \vec x_\perp\big) ,\, \widehat{\phi}\big(x^+, y^-, \vec y_\perp\big) \Big] \,=\, -\mathrm{i}\hbar \tfrac{1}{2} \delta\big(x^- - y^-\big) \delta^{d-1}\big(\vec x_\perp - \vec y_\perp\big) \mathrlap{\,.}$$

And so the above Schwinger-Dyson argument, just with the time coordinate $x^0$ replaced by the light front parameter $x^+$, reproduces this in the form:

(9)$$\begin{array}{l} \underset{\underset{\epsilon\to 0}{\longrightarrow}}{\lim} \Big\langle \pi\big(y^+ + \epsilon, x^-, \vec x_\perp\big) \, \phi\big(y^+, y^-, \vec y_\perp\big) \;-\; \pi\big(y^+ - \epsilon, x^-, \vec x_\perp\big) \, \phi\big(y^+, y^-, \vec y_\perp\big) \Big\rangle \\ \;=\; -\mathrm{i}\hbar \, \tfrac{1}{2} \delta(x^- - y^-) \delta^{d-1}(\vec x_\perp - \vec y_{\perp}) \mathrlap{\,.} \end{array}$$

(Just beware the somewhat subtle factor of $1/2$ on the right of (9). In the constrained canonical quantization this factor may be found discussed carefully in Burkardt 1996 §A p. 76. In the path integral picture the factor arises more transparently as a factor of $2$ on the left, originating in: $\delta(\partial_+ \phi \partial_- \phi)/\delta\phi = -\partial_+ \partial_- \phi - \partial_- \partial_+ \phi = -2 \partial_+ \partial_- \phi$.)