Quantum field theory 1, lecture 08

4.5 Functional integral for expectation values of time-ordered operators

We first derive the functional integral expression for the propagator. The corresponding technical steps are easily generalised to arbitrary chains of time-ordered operators.

Propagator from functional integral. For the derivation of a functional integral expression for the propagator we employ the functional integral expression for the evolution operator in the expression \begin{equation*} G(\vec{y},t_2;\vec{x},t_1) = \langle 0| U^\dagger (t_2,t_\text{in}) a(\vec{y}) U(t_2,t_1) a^\dagger (x) U(t_1,t_\text{in})|0\rangle . \end{equation*} One often calls \(|0\rangle = |0\rangle _\text{in}\) the initial vacuum at \(t_{\text{in}}\), and \(|0\rangle _{\text{f}} = U(t_f,t_\text{in})|0\rangle _\text{in}\) the final vacuum at \(t_{\text{f}}\). For a time-translation invariant vacuum one has \(|0\rangle _{\text{f}} = |0\rangle _{\text{in}}\). This implies \begin{equation*} \langle 0|_{\text{f}} = \langle 0| U^\dagger (t_{\text{f}},t_{\text{in}}). \end{equation*} Using \begin{equation*} U^\dagger (t_f,t_\text{in}) U(t_f,t_2) = U^\dagger (t_2, t_\text{in}), \end{equation*} we find \begin{equation*} G(\vec{y},t_2;\vec{x},t_1) = \langle 0|_{\text{f}} U(t_f,t_2) a(\vec{y}) U(t_2,t_1) a^\dagger (\vec x) U(t_1,t_\text{in})|0\rangle . \end{equation*} This intuitive expression for the propagator involves evolution operators that can be expressed in terms of the functional integral.

We have derived before the functional integral expression for the evolution operator \begin{equation*} U(t_2,t_1) = \int d x(t_2) \int d p(t_1) \, | x(t_2)\rangle \tilde{F}(t_2,t_1) \langle p(t_1)|, \end{equation*} with \begin{equation*} \tilde{F}(t_2,t_1)= \int D \varphi (t_1 < t' <t_2) \exp \left [-\int _{t_1}^{t_2} dt \mathscr{L}(t)\right ]. \end{equation*} Here \(|x(t) \rangle \) and \(|p(t) \rangle \) are eigenstates of the abstract operators \(\hat{x}\) and \(\hat{p}\) which are not related to positions in space and momenta, \begin{equation*} \hat{x}|x(t)\rangle = x(t)|x(t)\rangle ,\quad \quad \quad \hat{p}|p(t)\rangle = p(t)|p(t)\rangle . \end{equation*} We employ now a mixed basis with \(x\) and \(p\), which is reflected by the difference between \(\tilde{F}\) and \(F\) as used previously. The integrals over \(x(t_2)\) and \(p(t_1)\) are not yet included in \(\int D\phi (t_1 <t'< t_2).\)

Our expression for the propagator contains factors \begin{equation*} \begin{split} U(t_3,t_2) \, \hat{A} \, U(t_2,t_1) = & \int d x(t_3)\;d p(t_2)\;d x(t_2)\;d p(t_1) \\ &\times |x(t_3)\rangle \, \tilde{F}(t_3,t_2) \, \langle p(t_2)| \, \hat{A} \, |x(t_2)\rangle \, \tilde{F}(t_2,t_1) \, \langle p(t_1)|, \end{split} \end{equation*} for which we need the matrix element \begin{equation*} \langle p(t_2)| \, \hat{A} \, |x(t_2)\rangle = A(x(t_2),p(t_2)). \end{equation*}

For \(\hat{A}\) depending on \(a^\dagger \) and \(a\) we first express it in terms of the operators \(\hat{x}\) and \(\hat{p}\), recalling the relations

\begin{equation*} a=\frac{1}{\sqrt{2}}(\hat{x}+i\hat{p}),\quad \quad \quad a^\dagger = \frac{1}{\sqrt{2}}(\hat{x}-i\hat{p}). \end{equation*} We can then replace in the matrix element for \(\hat{A}\)

\begin{equation*} \begin{split} a & \to \frac{1}{\sqrt{2}}\left [x(t_2)+ip(t_2) \right ], \\ a^\dagger & \to \frac{1}{\sqrt{2}}\left [ x(t_2) - ip(t_2) \right ], \end{split} \end{equation*} provided that the ordering of operators is done such that \(\hat{x}\)-operators are on the right and \(\hat{p}\)-operators on the left.

With the matrix element replaced by a function \(A(x_2,p_2)\), we can combine the remaining pieces to

\begin{equation*} \begin{split} U(t_3,t_2) \, \hat{A} \, U(t_2,t_1) = & \int dx(t_3) dx(t_1) \, |x(t_3) \rangle \, \int D\varphi (t_1 < t'<t_3)\; \\ & \times \text{exp}\; \left \{-\int ^{t_3}_{t_1} dt' \mathscr{L}(t')\right \} \, A(x(t_2),p(t_2))\, \langle x(t_1)|. \end{split} \end{equation*} In summary we get the rule: The operator \(\hat{A}\) at \(t_2\) leads to the insertion of a function \(A(t_2)\) into the functional integral.

Recall the inverse: an observable \(A(t)\) in the functional integral results in the insertion of an operator \(\hat{A}\) in the chain of transfer matrices.

Discrete formulation. We have been here a bit vague with the precise choice of integrations. In a precise discrete formulation one replaces \begin{equation*} \langle x_{j+1}|e^{-i\Delta t \hat{H}}|x_j\rangle \quad \text{by}\quad \langle x_{j+1}|e^{-i\Delta t \hat{H}}\hat{A}|x_j\rangle \end{equation*} at the appropriate place in the chain.

Replacement rules We can now follow \(A(x(t_2),p(t_2))\) through the chain of variable transformations \begin{equation*} x_j \to \tilde{x}_n \to \frac{1}{\sqrt{2}}(\varphi _n + \varphi ^{*}_{-n}) \to \frac{1}{\sqrt{2}}(\varphi (t)+\varphi ^{*}(t)), \end{equation*} and similarly \begin{equation*} p_j \to \tilde{p}_n \to -\frac{i}{\sqrt{2}}(\varphi _n-\varphi ^{*}_{-n}) \to -\frac{i}{\sqrt{2}}(\varphi (t) - \varphi ^{*}(t)), \end{equation*} resulting in the simple replacement rules \begin{equation*} a \to \varphi (t),\quad \quad \quad a^\dagger \to \varphi ^{*}(t). \end{equation*}

Propagator. This yields for the propagator correlation function \begin{equation*} G(\vec{y}, t_2, \vec{x}, t_1) = Z^{-1} \int D\varphi \; e^{-S[\varphi ]} \varphi (\vec{y},t_2)\varphi ^{*}(\vec{x},t_1). \end{equation*}

Expectation values for complex functional integrals. For complex functional integrals in Minkowski space we define expectation values similar to classical statistical physics \begin{equation*} \langle A \rangle = Z^{-1}\int D\varphi \;e^{-S[\varphi ]}A[\varphi ] \end{equation*} \begin{equation*} Z = \int D\varphi \; e^{-S[\varphi ]}. \end{equation*} With this one can write the propagator as \begin{equation*} G(\vec{y}, t_2, \vec{x}, t_1) = \langle \varphi (\vec{y},t_2) \varphi ^{*}(\vec{x},t_1) \rangle , \end{equation*} which is also known as the two-point correlation function.

Origin of the normalization factor \(Z\). We have not paid much attention to the normalization of the wave function, the additive normalization of the action, and the formal boundary terms. All this is accounted for by \(Z^{-1}\).

Fourier space. Since \(A[\varphi ]\) is a function (functional) of \(\varphi \), variable transformations are straightforward. No complications with commutator relations as for \(a, a^\dagger \). The Fourier transform of correlation function reads \begin{equation*} G(\vec{q}, t_2; \vec{p},t_1) = \int _y \int _x e^{-i\vec{q}\cdot \vec{y}}\;e^{i\vec{p} \cdot \vec{x}}\; G(\vec{y},t_2; \vec{x},t_1). \end{equation*} Translation symmetry implies \begin{equation*} G \sim \delta (\vec{q}-\vec{p}). \end{equation*}

Non-trivial field expectation values. So far we have assumed implicitly that the vacuum is trivial. In general \(\langle \varphi (\vec{x},t)\rangle \) may be different from zero. A more general definition of the (connected) correlation function is given by \begin{equation*} G(\vec{y},t_2; \vec{x},t_1)= \langle \delta \varphi (\vec{y},t_2) \delta \varphi (\vec{x},t_1)\rangle ,\quad \quad \quad \delta \varphi = \varphi -\langle \varphi \rangle . \end{equation*}

Definition of quantum field theory. A quantum field theory is defined by

  • Choice of fields \(\varphi \)
  • Action as functional of fields \(S[\varphi ]\)
  • Measure \(\int D\varphi \)

Correlation function. The correlation function is defined by \begin{equation*} G_{\alpha \beta } = \langle \varphi _\alpha \varphi ^{*}_\beta \rangle -\langle \varphi _\alpha \rangle \langle \varphi ^{*}_\beta \rangle , \end{equation*} with \(\alpha ,\beta \) collective indices, e.g. \(\alpha = (\vec{x},t)\) or \((\vec{p},t)\). No need of knowledge of vacuum. This is important, since the precise properties of the vacuum for interacting theories are not known.

Chains of operators. Consider for \(t_n>t_{n-1}>.....t_2>t_1\) a chain of Heisenberg operators \begin{equation*} \tilde{G} = \langle 0| \hat{A}^{(n)}_H(t_n)\hat{A}_H^{(n-1)}(t_{n-1})\ldots \hat{A}^{(2)}_H(t_2) \hat{A}^{(1)}_H(t_1)|0\rangle \end{equation*} The Green’s function is a special case \begin{equation*} G = \langle 0| a_H(t_2) a^\dagger _H(t_1)|0\rangle . \end{equation*} In complete analogy one finds the functional integral expression \begin{equation*} \tilde{G} = Z^{-1}\int \;D\varphi \;e^{-S}\bar{A}= \langle A \rangle \end{equation*} for the observable \begin{equation*} \bar{A} = A(t_n) A(t_{n-1})\cdots A(t_2) A(t_1) \end{equation*} with \begin{equation*} A(t_n) = A(\varphi ^{*}(t_n), \varphi (t_n)). \end{equation*}

Time ordering. The product \(A(t') A(t) = A(t) A(t')\) is commutative. The product \(\hat{A}_H(t') \hat{A}_H(t)\) in general not. What happens to commutation relations?

Define the time order operator \(T\) by putting in a product of two Heisenberg operators the one with larger time argument to the left. e.g. for \(t_2>t_1\), \begin{equation*} \begin{split} &T\left (\hat{A}^{(2)}_H(t_2) \hat{A}^{(1)}_H(t_1) \right ) = \hat{A}^{(2)}_H(t_2) \hat{A}^{(1)}_H(t_1)\\ &T\left (\hat{A}^{(1)}_H(t_1) \hat{A}^{(2)}_H(t_2) \right ) = \hat{A}^{(2)}_H(t_2) \hat{A}^{(1)}_H(t_1). \end{split} \end{equation*} The time ordered operator product is commutative. Generalize to products with several factors. \begin{equation*} \langle 0|T\left (\hat{A}_H \right )|0\rangle = \langle A \rangle \end{equation*} On the left one has an operator expression, and on the right functional integral expression.

Transition amplitude for multi-particle states. Consider two particles at \(t_1\) with momenta \(\vec{p}_1\) and \(\vec{p}_2\), and compute the transition amplitude to a two particle state at \(t_2 > t_1\) with momenta \(\vec{p}_3\) and \(\vec{p}_4\), \begin{equation*} \begin{split} \tilde{G}_{2,2} &= \langle 0| a_H(\vec{p}_4, t_2) a_H(\vec{p}_3, t_2) a^\dagger _H(\vec{p}_2,t_1) a^\dagger _H(\vec{p}_1,t_1)|0\rangle \\ &= \langle \varphi (\vec{p}_4,t_2) \varphi (\vec{p}_3,t_2) \varphi ^{*}(\vec{p}_2, t_1) \varphi ^{*}(\vec{p}_1,t_1)\rangle . \end{split} \end{equation*} This is a four-point function. It is a basic element of scattering theory.