nLab Weyl relation

Idea

What are called Weyl relations is the incarnation of canonical commutation relations under passing to exponentials, constituting the Weyl algebra.

For example if $a, a^\ast$ are two elements of an associative algebra with commutator

$$[a,a^\ast] = \hbar$$

then the corresponding Weyl relation is, by the Baker-Campbell-Hausdorff formula,

$$e^{z a} e^{z^\ast a^\ast} \;=\; e^{z^\ast a^\ast} e^{z a} e^{\hbar z z^\ast}$$

for $z,z^\ast \in \mathbb{C}$.

In the Wick algebra of free quantum fields

Proposition

(Hadamard-Moyal star product on microcausal observables – abstract Wick algebra)

Let $(E,\mathbf{L})$ a free Lagrangian field theory with Green hyperbolic equations of motion $P \Phi = 0$. Write $\Delta$ for the causal propagator and let

$$\Delta_H \;=\; \tfrac{i}{2}\Delta + H$$

be a corresponding Wightman propagator (Hadamard 2-point function).

Then the star product induced by $\Delta_H$

$$A \star_H A \;\coloneqq\; prod \circ \exp\left( \int_{X^2} \hbar \Delta_H^{a b}(x_1, x_2) \frac{\delta}{\delta \Phi^a(x_1)} \otimes \frac{\delta}{\delta \Phi^b(x_2)} dvol_g \right) (P_1 \otimes P_2)$$

on off-shell microcausal observables $A_1, A_2 \in \mathcal{F}_{mc}$ is well defined in that the wave front sets involved in the products of distributions that appear in expanding out the exponential satisfy Hörmander's criterion.

Hence by the general properties of star products (this prop.) this yields a unital associative algebra structure on the space of formal power series in $\hbar$ of off-shell microcausal observables

$$\left( PolyObs(E)_{mc}[ [\hbar] ] \,,\, \star_H \right) \,.$$

This is the off-shell Wick algebra corresponding to the choice of Wightman propagator $H$.

Moreover the image of $P$ is an ideal with respect to this algebra structure, so that it descends to the on-shell microcausal observables to yield the on-shell Wick algebra

$$\left( PolyObs(E,\mathbf{L})_{mc}[ [ \hbar ] ] \,,\, \star_H \right) \,.$$

Finally, under complex conjugation $(-)^\ast$ these are star algebras in that

$$\left( A_1 \star_H A_2 \right)^\ast = A_2^\ast \star_H A_1^\ast \,.$$

For proof see at Wick algebra this prop..

Remark

(Wick algebra is formal deformation quantization of Poisson-Peierls algebra of observables)

Let $(E,\mathbf{L})$ a free Lagrangian field theory with Green hyperbolic equations of motion $P \Phi = 0$ with causal propagator $\Delta$ and let $\Delta_H \;=\; \tfrac{i}{2}\Delta + H$ be a corresponding Wightman propagator (Hadamard 2-point function).

Then the Wick algebra $\left( PolyObs(E,\mathbf{L})_{mc}[ [\hbar] ] \,,\, \star_H \right)$ from prop. is a formal deformation quantization of the Poisson algebra on the covariant phase space given by the on-shell polynomial observables equipped with the Poisson-Peierls bracket $\{-,-\} \;\colon\; PolyObs(E,\mathbf{L})_{mc} \otimes PolyObs(E,\mathbf{L})_{mc} \to PolyObs(E,\mathbf{L})_{mc}$ in that for all $A_1, A_2 \in PolyObs(E,\mathbf{L})_{mc}$ we have

$$A_1 \star_H A_2 \;=\; A_1 \cdot A_2 \;mod\; \hbar$$

and

$$A_1 \star_H A_2 - A_2 \star_H A_1 \;=\; i \hbar \{A_1, A_2\} \;mod\; \hbar^2 \,.$$

(Dito 90, Dütsch-Fredenhagen 01)

Proof

By prop. this is immediate from the general properties of the star product (this example).

Explicitly, consider, without restriction of generality, $A_1 = \int (\alpha_1)_a(x) \mathbf{\Phi}^a(x)\, dvol_\Sigma(x)$ and $A_2 = \int (\alpha_2)_a(x) \mathbf{\Phi}^a(x)\, dvol_\Sigma(x)$ be two linear observables. Then

$$\begin{aligned} A_1 \star_H A_2 & = A_1 A_2 + \hbar \int \left( \tfrac{i}{2} \Delta^{a_1 a_2}(x_1, x_2) + H^{a_1 a_2}(x_1,x_2) \right) \frac{\partial A_1}{\partial \mathbf{\Phi}^{a_1}(x_1)} \frac{\partial A_2}{\partial \mathbf{\Phi}^{a_2}(x_2)} \;mod\; \hbar^2 \\ & = A_1 A_2 + \hbar \left( \int (\alpha_1)_{a_1}(x_1) \left( \tfrac{i}{2}\Delta^{a_1 a_2}(x_1, x_2) + H^{a_1 a_2}(x_1, x_2) \right) (\alpha_2)_{a_2}(x_2) \right) \;mod\; \hbar^2 \end{aligned}$$

Now since $\Delta$ is skew-symmetric while $H$ is symmetric is follows that

$$\begin{aligned} A_1 \star_H A_2 - A_2 \star_H A_1 & = i \hbar \left( \int (\alpha_1)_{a_1}(x_1) \Delta^{a_1 a_2}(x_1, x_2) (\alpha_2)_{a_2}(x_2) \right) \;mod\; \hbar^2 \\ & = i \hbar \, \left\{ A_1, A_2\right\} \end{aligned} \,.$$

The right hand side is the integral kernel-expression for the Poisson-Peierls bracket, as shown in the second line.

Example

(Hadamard vacuum state 2-point function)

Let

$$A_i \in LinObs(E,\mathbf{L})_{mc} \hookrightarrow PolyObs(E,\mathbf{L})_{mc}$$

for $i \in \{1,2\}$ be two linear microcausal observables represented by distributions which in generalized function-notation are given by

$$A_i \;=\; \int (\alpha_i)_{a_i}(x_i) \mathbf{\Phi}^{a_i}(x_i) \, dvol_\Sigma(x_i) \,.$$

Then their Hadamard-Moyal star product (prop. ) is the sum of their pointwise product with $\tfrac{1}{2} i \hbar$ times the evaluation

$$\begin{aligned} \langle A_1 A_2\rangle & \coloneqq \int \int (\alpha_1)_{a_1}(x_1) \, \left\langle \mathbf{\Phi}^{a_1}(x_1) \mathbf{\Phi}^{a_2}(x_2)\right\rangle \, (\alpha_2)_{a_2}(x_2) \, dvol_\Sigma(x_1) \, dvol_\Sigma(x_2) \\ & \coloneqq \tfrac{1}{2} i \hbar \int \int (\alpha_1)_{a_1}(x_1) \Delta_H^{a_1 a_2}(x_1,x_2) (\alpha_2)_{a_2}(x_2) \,dvol_\Sigma(x_1) \,dvol_\Sigma(x_2) \end{aligned}$$

of the Wightman propagator $\Delta_H$:

(1)$$A_1 \star_H A_2 = A_1 \cdot A_2 + \langle A_1 A_2\rangle$$

Further below we see that this evaluation is the 2-point function of a state on the Wick algebra.

Example

(Weyl relations)

Let $(E,\mathbf{L})$ a free Lagrangian field theory with Green hyperbolic equations of motion and with Wightman propagator $\Delta_H$.

Then for

$$A_1, A_2 \;\in\; LinObs(E,\mathbf{L})_{mc} \hookrightarrow PolyObs(E,\mathbf{L})_{mc}$$

two linear microcausal observables, the Hadamard-Moyal star product (def. ) of their exponentials exhibits the Weyl relations:

$$e^{A_1} \star_H e^{A_2} \;=\; e^{A_1 + A_2} \; e^{\langle A_1 A_2\rangle}$$

where on the right we have the exponential Wightman 2-point function (1).

(e.g. Dütsch 18, exercise 2.3)

Related concepts

• Weyl algebra

• canonical commutation relations

• Heisenberg algebra

References

For more references see at Weyl algebra.

The notion goes back to

• Hermann Weyl, (46) in: Quantenmechanik und Gruppentheorie, Zeitschrift für Physik 46 (1927) 1–46 [doi:10.1007/BF02055756]

• John von Neumann, p. 571 of: Die Eindeutigkeit der Schrödingerschen Operatoren, Mathematische Annalen 104 (1931) 570–578 [doi:10.1007/BF01457956]

  (proving the Stone-von Neumann theorem)

See also:

• Michael Dütsch, exercise 2.3 in: From classical field theory to perturbative quantum field theory, 2018