37.3 The propagator in non-relativistic quantum field theory

In quantum field theory the Heisenberg picture operators that provide observables will be products of the field operators, and the time-dependence of these for the free-particle theory was determined in section 37.2. For the timeindependent state, the natural choice is the vacuum state |0⟩, although other possibilities such as coherent states may also be useful. States with a finite number of particles will be given by applying field operators to the vacuum, so such states just corresponds to a diferent product of field operators.

We will not enter here into details, but a standard topic in quantum field theory textbooks is “Wick’s theorem”, which says that the calculation of expectation values of products of field operators in the state |0⟩ can be reduced to the problem of calculating the following special case:

Definition (Propagator for non-relativistic quantum field theory)

The propagator for a non-relativistic quantum field theory is the amplitude, for $t_{2} > t_{1}$
$U (x_{2}, t_{2}, x_{1}, t_{1}) = ⟨ 0∣ Ψ (x_{2}, t_{2}) Ψ^{†} (x_{1}, t_{1}) ∣0 ⟩$

The physical interpretation of these functions is that they describe the amplitude for a process in which a one-particle state localized at $x_{1}$ is created at time $t_{1} .$ , propagates for a time $t_{2} - t_{1}$ , and is annihilated at position $x_{2}$ . Using the solution for the time-dependent field operator given earlier we find

U (x_{2}, t_{2}, x_{1}, t_{1}) = \frac{1}{2 π} \iint_{R^{2}} ⟨ 0∣ e^{i p_{2} x_{2}} e^{- i \frac{p _{2}^{2}}{2 m} t} a (p_{2}) e^{- i p_{1} x_{1}} e^{i \frac{p _{1}^{2}}{2 m} t_{1}} a^{†} (p_{1}) ∣0 ⟩ d p_{2} d p_{1} = \frac{1}{2 π} \iint_{R^{2}} e^{i p_{2} x_{2}} e^{- i \frac{p _{2}^{2}}{2 m} t_{2}} e^{- i p_{1} x_{1}} e^{i \frac{p _{1}^{2}}{2 m} t_{1}} δ (p_{2} - p_{1}) d p_{2} d p_{1} = \frac{1}{2 π} \int_{- \infty}^{+ \infty} e^{- i p (x_{1} - x_{2})} e^{- i \frac{p ^{2}}{2 m} (t_{2} - t_{1})} d p

This is exactly the same calculation (see equation 12.5) already discussed in detail in section 12.5. As described there, the result (equation 12.9) is

U (x_{2}, t_{2}, x_{1}, t_{1}) = U (t_{2} - t_{1}, x_{2} - x_{1}) = (\frac{m}{i 2 π ( t _{2} - t _{1} )})^{\frac{1}{2}} e^{\frac{m}{2 i ( t _{2} - t _{1} )} (x_{2} - x_{1})^{2}}

which satisfies

t \to 0^{+} lim U (t, x_{2} - x_{1}) = δ (x_{2} - x_{1})

If we extend the definition of $U (t, x_{2} - x_{1})$ to $t < 0$ by taking it to be zero there, as in section we get the retarded propagator $U_{+} (t, x_{2} - x_{1})$ and its Fourier transformed version in frequency-momentum space of section 12.6 as well as the relation to Green’s functions of section 12.7.

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正文：英文 · OCR 机器稿 · 待校对

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原书 PDF · 印刷页 394、395、396、397、398、399、400、401、402、403

来源版本：2025-10-20

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37.3 The propagator in non-relativistic quantum field theory

37.3 The propagator in non-relativistic quantum field theory

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