In quantum field theory one usually calculates the Feynman propagator defined as the time ordered product of (scalar) fields: $$iG_F(x,x')=\langle0\lvert T[\phi(x)\phi(x')]\rvert0\rangle \tag{1}$$

However sometimes you come across other vacuum expectation values (VEV), especially for the commutator and anticommutators, denoted respectively by: $$iG(x,x')=\langle0\lvert [\phi(x),\phi(x')] \rvert 0\rangle \tag{2}$$ $$G^{(1)}(x,x')=\langle0\lvert\{\phi(x),\phi(x') \} \rvert 0\rangle \tag{3},$$ where $G(x,x')$ is known as Schwinger function and $G^{(1)}(x,x')$ as Hadamard elementary function.

Now if I've understood correctly, the VEV of the time ordered product $(1)$ describes the propagation of a particle from $x'$ to $x$ if $t_{x'}<t_x$ or the other way around.

The question is that is there a same type of physical interpretation for the two other VEV of equations $(2)$ and $(3)?$


1 Answer 1


Well, "physical meaning" is a quit ambiguous word, actually.

What one can say is that, regarding so-called Gaussian states as Minkowski quantum vacuum, the Wightman two-point function $w_2(x,y) = \langle \psi|\phi(x) \phi(y) \psi \rangle$ permits one to reconstruct both the state and the representation of the whole theory in a Hilbert space, up to isomorphisms.

This result is a subcase of a more general result (see Strater-Wightman textbook for instance) regarding general states and the whole class of $n$-point functions $w(x_1, \ldots, x_n)$, $n=1,2,\ldots$.

Regarding $w_2(x,y)$ actually the quantum information is completely encompassed by its symmetric part, that is the Hadamard function you mention, since the anti-symmetric part does not distinguish among different quantum states as it satisfies

$$\frac{1}{2} \langle \psi|(\phi(x) \phi(y) - \phi(y) \phi(x))\psi \rangle = \frac{i}{2}E(x,y) \langle \psi| I \psi\rangle = \frac{i}{2}E(x,y) $$ where $E$ is the so-called causal propagator, the difference of the advanced Green function, $A$, and the retarded, one $R$, of the Klein-Gordon operator. $A$ and $R$ are completely classical (non-quantum) objects: They do not need any quantization procedure to be defined, just the classical theory of hyperbolic PDEs.

The Schwinger function essentually is $E$ above defined. $E$ is a map associating smooth compactly supported functions $f$ with solutions of (homogeneous) KG equation $\psi_f (x) = \int E(x,y) f(y) dy$ supported in $J^+(supp f) \cup J^-(supp f)$. The existence of $E$ permits to define field operators smeared with solutions instead of compactly supported functions. This opportunity is useful in sevearl points in developing the theory, for instance the LSZ formalism.


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