Quantum field theory 1, lecture 07

4.4 Real time evolution

The functional integral can also be employed for the time evolution of quantum systems. This is typically a problem with boundary conditions. An initial condition for the quantum state is given at some initial time \(t_{in}\) by the wave function \(\psi (t_\text{in})\). This wave function develops in time according to the unitary evolution in quantum mechanics and arrives at some final time \(t_f\) at \begin{equation*} \psi (t_f)=U(t_f,t_{in}) \psi (t_{in}). \end{equation*} For a time-independent Hamiltonian \(H\) the evolution operator obeys \begin{equation*} U(t_f,t_{in})=U(t_f-t_{in}) = e^{-i(t_f - t_{\text{in}}) H}. \end{equation*} We are interested in the transition amplitude to some different final wave function \(\phi (t_f)\).

We want to derive the functional integral for the transition amplitude \begin{equation*} \langle \phi (t_f)| \psi (t_f)\rangle = \langle \phi (t_f)| U(t_f-t_{\text{in}})|\psi (t_\text{in})\rangle = \langle \phi (t_f)| e^{-i(t_f - t_{\text{in}})H}|\psi (t_\text{in})\rangle . \end{equation*} Recalling our formulation of thermal equilibrium with boundary conditions and its extension to a complex formulation, the transition element can be obtained from thermal equilibrium by the replacement \begin{equation*} \beta \to i(t_f - t_{\text{in}}). \end{equation*} The split into infinitesimal pieces, Fourier-transforms etc can be done for complex \(\beta \) in the same way as before. For \(\beta \to \infty ~ (T \to 0),~t_f-t_{\text{in}}\to \infty \) one finds \begin{equation*} \langle \phi (t_f)|\psi (t_f)\rangle = B(t_f,t_{\text{in}})Z_t \end{equation*} \begin{equation*} Z_t = \int D\phi \; \text{exp}(-S_t). \end{equation*} For the sake of clarity we denote by \(S_t\) the action for the dynamical time evolution, in contrast to \(S_{eq}\) for the thermal equilibrium. For obtaining \(S_t\) from \(S_{eq}\) we have to multiply the terms \(\sim \beta \) by \(i\) before taking the limit \(\beta \longrightarrow \infty .\) The term \(i\tilde \omega \) remains unchanged, while all other parts in the action get multiplied by \(i\). This results in \begin{equation*} \begin{split} S_t &= \int _q \Bigg [\phi ^*(q)\left [i \tilde{\omega } + i \left (\frac{\vec{q}^2}{2M}-\mu +\lambda \delta \mu \right )\right ]\phi (q) \\ &\qquad + i \frac{\lambda }{2}\int _{q_1}\int _{q_2}\int _{q_3}\int _{q_4} \phi ^*(q_4)\phi ^*(q_3)\phi (q_2)\phi (q_1)\delta (q_3 + q_4-q_1-q_2)\Bigg ]. \end{split} \end{equation*}

After a Fourier-transform in \(\tilde{\omega }\) and \(\vec{q}\) one finds, with time labelled now by \(t\) \begin{equation} S_t = \int _x[\phi ^*(x) \partial _t \phi (x) + \frac{i}{2M}(\vec{\nabla }\phi ^*(x))(\vec{\nabla }\phi (x)) + \frac{i\lambda }{2}(\phi ^*(x)\phi (x))^2 -(\mu -\lambda \delta \mu ) \phi ^{*}(x) \phi (x)] \label{eq:actionEq} \end{equation} where \begin{equation*} x= (t,\vec{x}), \quad \quad \quad \int _x = \int _{-\infty }^\infty dt \int d^3 \vec{x}. \end{equation*} The transfer matrix for this functional integral is now \begin{equation*} \hat{T_t}=exp\left [-\frac{i(t_f-t_\text{in})}{(2N+1)}H\right ] \end{equation*} instead of \begin{equation*} \hat{T_{eq}} = exp\left [-\frac{\beta }{(2N+1)}H\right ]. \end{equation*} The matrix \(\hat{T}_t\) is a unitary matrix if the Hamiltonian is hermitean, \(H^\dagger = H.\)

Local Physics For observations and experiments done in some time involved around \(t\) the details of boundary conditions at \(t_f\) and \(t_{\text{in}}\) play no role for large \(|{t_f - t}|\) and \(|{t-t_{\text{in}}}|.\) Doing physics now is not much influenced by what happened precisely to the dinosaurs or what will happen in the year 10000. For many purposes the boundary term \(B(t_f,t_{\text{in}})\) is just an irrelevant multiplicative factor in \(Z\) which drops out from the expectation values of interest. One can then simply omit it and work directly with \(Z_t\).

Minkowski action. We define the Minkowski action \(S_M\) by multiplying the euclidean action \(S\) with a factor \(i\) \begin{equation*} S_M = i S, \quad \quad \quad e^{-S} = e^{i S_M}. \end{equation*} This can be done both for \(S_t\) and \(S_{eq}\). For \(S_t\) the Minkowski action reads \begin{equation*} S_{M,t} = -\int _x \phi ^*\left (-i\partial _t - \frac{\Delta }{2M}\right )\phi + \frac{\lambda }{2}(\phi ^*(x)\phi (x))^2+ \dots . \end{equation*} Variation of \(S_{M,t}\) or \(S_t\) with respect to \(\phi ^*\) yields for \(\lambda =0\) the Schrödinger equation for the wave function of a single free particle \begin{equation*} \left (-i\partial _t - \frac{\Delta }{2M} \right )\phi = 0. \end{equation*}

There is a reason for that, but the connection needs a few steps, concentrating on single particle states. Recall that the functional integral describes arbitrary particle numbers, such that one-particle states are only special cases. For \(\lambda \neq 0 \) the classical field equation \(\frac{\delta S}{\delta \phi ^*(x)}=0\) is a non-linear equation, called Gross-Pitaevskii equation \begin{equation*} i\partial _t \phi = -\frac{\Delta }{2M}\phi + \lambda (\phi ^*\phi )\phi - (\mu -\lambda \delta \mu ) \varphi \end{equation*} This is not a linear Schrödinger equation for a quantum wave function, but has a different interpretation. An equation of this type can describe the dynamics of a Bose-Einstein condensate.

Analytic continuation. Let us replace in the action \(S_t\) \eqref{eq:actionEq} the time coordinate \begin{equation*} t= - i\tau \end{equation*} such that the integration becomes \begin{equation*} \int _x = -i \int d\tau d^3 \vec{x}. \end{equation*} For the time derivative term we have \begin{equation*} \partial _t \phi = i\partial _\tau \phi . \end{equation*} This replacement is called ”analytic continuation”. The analytic continuation of \(S_t\) is the action \(S_{eq}\) for thermal equilibrium at zero temperature, \begin{equation*} S_t \to S_{eq}= \int d\tau d^3 x \left \{ \phi ^*\left (\partial _\tau - \frac{\Delta }{2M}-\mu \right )\phi + \frac{\lambda }{2}(\phi ^*\phi )^2 + \lambda \delta \mu \phi ^{*}\phi \right \}. \end{equation*} Analytic continuation can be done in both ways. The actions \(S_t\) and \(S_{eq}\) for two models, one for the time evolution, the other for the \(T=0\) limit of thermal equilibrium, are related by analytic continuation.

Note that \(S_M\) is not the analytic continuation of \(S\), but rather related to \(S\) by a fixed definition. The sign of \(S_M\) is of historical origin. The Minkowski action \(S_{M,t}\) is a real expression. In consequence, \(e^{i S_{M,t}}\) is a phase. This is a profound change as compared to the situation for thermal equilibrium, for which \(e^{i S_{eq}} = e^{-S_{eq}}\) is a positive real quantity that can be associated to a probability distribution. The functional integral for the time evolution of quantum systems is described by an integration over phases. This is directly related to the unitary evolution in quantum mechanics. The transfer matrix \(\hat T_t\) is a unitary matrix. No boundary information is lost, in contrast to the thermal equilibrium state, for which we have seen for the Ising chain how the memory of boundary information is lost in the bulk.

Fourier transform. For the Fourier transformation into frequency space we employ \begin{equation*} \tilde{\omega }\tau = \omega _M t = -i\omega _M \tau \end{equation*} This defines the Minkowski frequency \begin{equation*} \omega _M = i\tilde{\omega } = q_0 \end{equation*} where \(q_0\) is the zero-component of the four-momentum \(q_\mu \). Analytic continuation in time translates to analytic continuation in frequency or four-momentum between \(\tilde \omega \) and \(\omega _M\).

The analytic continuation in momentum space is a very useful tool for the evaluation of correlation functions. One can first compute the correlation functions in ”euclidean space”, which corresponds to the \(T \to 0\) limit of thermal equilibrium. This has the advantage that powerful methods can be used as, for example, numerical simulations. The correlation functions in momentum space depend on \(\tilde \omega \). Subsequently, they can be continued analytically to Minkowski space, with replacement rules for the frequencyies \begin{equation*} \tilde{\omega }\to -i q_0. \end{equation*} For the squared frequency one finds, using the Minkowski metric for raising and lowering indices, \(\eta _{00}=-1\), \begin{equation*} \tilde{\omega }^2 \to -(q_0)^2 = -(q^0)^2 = q^0 q^0 \eta _{00} = q^0 q_0. \end{equation*} For a relativistic theory one has \begin{equation*} q_E^2= \tilde{\omega }^2 + \vec{q}^2 \to q^0q_0 + q^iq_i = q^\mu q_\mu = - (q^0)^2 + \vec q^2= q_M^2, \end{equation*} and analytic continuation corresponds to \begin{equation*} q_E ^2 \to q_M^2 \end{equation*}

For a vacuum state with translation and rotation symmetry the two-point correlation function can only depend on the invariant \(q^2=q^{\mu } q_{\mu }\). Only the meaning of \(q^2=q^{\mu } q_{\mu }\) differs between euclidean and Minkowski signature. For euclidean signature the zero-index is lowered by \(\delta _{00}\), while for Minkowski signature one employs \(\eta _{00}\). Thus analytic continuation can also be formulated as an analytic continuation in the metric. For euclidean signature \(q^2\) is invariant under \(SO(4)\)-rotations in four-dimensional euclidean space, while for Minkowski signature the Lorentz symmetry \(SO(1,3)\) leaves \(q^2\) invariant. Many properties can be understood by viewing momenta in the complex plane, for which analytic continuation can be formulated as a continuous rotation of \(q_0\).

4.5 Expectation values of time ordered operators

So far we have established for the partition function a map between the operator formalism and the functional integral. This extends to the expectation values of observables. For the functional integral formulation expectation values are directly found by inserting the observable in the functional integral. An observable is a functional of the fields for which the functional integral is formulated. It is a c-number and no non-commuting structures are present at this level. The definition of the expectation value of an observable \(A[\phi ]\) holds independently of the particular form of the action, \begin{equation*} \langle A \rangle = Z^{-1} \int D\phi e^{-S[\phi ]} A[\phi ]. \end{equation*} In particular, it is valid both for euclidean and Minkowski signature of the metric.

We have seen in sect.3.2 how operators can be associated to observables. They allow us to express expectation values in the functional integral by time-ordered products of Heisenberg operators. We will next establish the inverse direction and show how the expectation values of time ordered operators in the operator formalism translate to the functional integral expression. At the stage where we are this should no longer be a surprise. Nevertheless, we perform this step here, repeating partly the construction of the functional integral from the operator formalism. This provides for a link to many textbooks where the functional integral expression in introduced in this way.

Heisenberg picture in quantum mechanics. We briefly recapitulate the Heisenberg picture in quantum mechanics. While in the Schrödinger picture the wave function evolves and the operators are constant, in the Heisenberg picture the operators evolve instead. The central objects are \(\hat{A}_H(t)\), the Heisenberg operators that depend on time. One can write them as \begin{equation*} \hat{A}_H(t) = U^\dagger (t, t_\text{in}) \hat{A}_S U(t,t_\text{in}), \end{equation*} where \(\hat A_S\) is the operator in the Schrödinger picture. Consider for \(t_2 \geq t_1\) \begin{equation*} \hat{A}_H(t_2) \hat{B}_H(t_1) = U^\dagger (t_2,t_\text{in}) \, \hat{A}_S \, U(t_2,t_\text{in}) U^\dagger (t_1, t_\text{in}) \, \hat{B}_S \, U(t_1, t_\text{in}), \end{equation*} and use \begin{equation*} U^\dagger (t_1, t_2) = U(t_2, t_1), \end{equation*} as well as \begin{equation*} U(t_3,t_2) U(t_2,t_1) = U(t_3,t_1). \end{equation*} With \begin{equation*} U(t_2,t_{\text{in}}) U^\dagger (t_1,t_\text{in}) = U(t_2,t_1) U(t_1,t_\text{in}) U^\dagger (t_1,t_\text{in}) = U(t_2,t_1), \end{equation*} one has \begin{equation*} \hat{A}_H (t_2) \hat{B}_H (t_1) = U^\dagger (t_2, t_\text{in}) \, \hat{A}_S \, U(t_2,t_1) \, \hat{B}_S \, U(t_1, t_\text{in}). \end{equation*} In the Heisenberg picture, one keeps the wave function fixed \(|\psi \rangle = |\psi (t_\text{in})\rangle \) and describes the time evolution by the time-dependence of the Heisenberg operators.

The transition amplitude for two time-ordered Heisenberg operators, where the larger t-argument stands on the left, is defined by \begin{equation*} \langle \phi (t_\text{in})| \hat{A}_H (t_2) \hat{B}_H(t_1)|\psi (t_\text{in})\rangle = \langle A(t_2) B(t_1)\rangle _{\phi \psi }. \end{equation*} It reads in the Schrödinger picture \begin{equation*} \begin{split} \langle A(t_2) B(t_1) \rangle _{\phi \psi } &= \langle \phi (t_\text{in})| U^\dagger (t_2,t_\text{in}) \,\hat{A}_S \, U(t_2,t_1) \, \hat{B}_S \, U(t_1, t_\text{in}) | \psi (t_\text{in})\rangle \\ &= \langle \phi (t_2)| \, \hat{A}_S \, U(t_2, t_1) \, \hat{B}_S \, | \psi (t_1)\rangle . \end{split} \end{equation*} We may insert a complete set of states \begin{equation*} \int d\chi (t_1)|\chi (t_1)\rangle \langle \chi (t_1)| = \mathbb{1}, \end{equation*} in order to obtain \begin{equation*} \begin{split} \langle A(t_2) B(t_1)\rangle _{\varphi \psi } &= \int d\chi (t_1) \langle \varphi (t_2) | \, \hat{A}_S \, U(t_2,t_1) | \chi (t_1)\rangle \langle \chi (t_1) | \, \hat{B}_S \, |\psi (t_1)\rangle \\ & = \int d\chi (t_1) \langle \varphi (t_2) | \, \hat{A}_S \, | \chi (t_2)\rangle \langle \chi (t_1) | \, \hat{B}_S \, | \psi (t_1)\rangle \end{split} \end{equation*} This has an intuitive interpretation: The transition amplitudes are evaluated for \(B\) at time \(t_1\) between \(\psi (t_1)\) and arbitrary intermediate states \(\chi (t_1).\) Then \(\chi (t_1)\) propagates in time to \(\chi (t_2),\) and one evaluates the transition amplitude at \(t_2\) of \(A\) between \(\chi (t_2)\) and \(\phi (t_2).\) One finally sums over intermediate states.

It is our aim to derive a functional integral expression for this transition amplitude. We will do this first for a particular amplitude, namely the propagator. This will then be generalised to arbitrary chains of time-ordered operators.

4.6 Propagator

The propagator is a central quantity in quantum field theory. It contains the information how a one-particle wave function at \(t_1\) has evolved at some later time \(t_2\). We will express the propagator as a suitable transition amplitude for a product of annihilation and creation operators. In the functional integral formalism it will be given by a connected two-point function.

Since the propagator deals with the dynamics of a single particle we first define basis functions for localised single particle states. Particles are excitations of the vacuum. We therefore start at \(t_{in}\) with an initial vacuum state \(|0\rangle \), evolve it to \(t_1\), and apply a creation operator \(a^\dagger (\vec{x})\). The result is a state for which at \(t_1\) a single particle is located precisely at \(\vec{x}\). We denote this one-particle state by \begin{equation*} a^\dagger (\vec{x}) U(t_1, t_\text{in})|0\rangle = |(\vec{x},t_1); t_1\rangle . \end{equation*}

Propagator as transition amplitude. For \(t>t_1\) the particle will move. Correspondingly, the wave function changes in the Schrödinger picture according to the standard evolution in quantum mechanics, \begin{equation*} |(\vec{x},t_1);t \rangle = U(t,t_1) |(\vec{x},t_1);t_1\rangle . \end{equation*} One has to distinguish the two different time arguments. For \(|(\vec{x},t_1); t \rangle \) the time argument \(t_1\) is a label (together with \(\vec{x}\)) specifying which state is meant. This state is the one for which at \(t_1\) the particle is located at \(\vec{x}\). The time argument in the Schrödinger evolution of this wave function is given by \(t\). For a given basis state \(t_1\) is kept fixed, while the time evolution of the wave function in the Schrödinger picture is the evolution with varying \(t\).

Let us define the transition amplitude of this one-particle state with a different one particle state \(|(\vec{y}, t_2); t \rangle \) at a given time \(t\). Its square is the probability to find a particle that was at time \(t_1\) at \(\vec{x}\) to be a particle that is at \(\vec{y}\) at time \(t_2\). This transition amplitude is the propagator, \begin{equation*} G(\vec{y},t_2; \vec{x},t_1) = \langle (\vec{y},t_2); t | (\vec{x}, t_1); t \rangle . \end{equation*}

The propagator can be expressed by a product of Heisenberg operators. For this purpose we take \(t=t_2\), \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 (\vec{x}) U(t_1,t_\text{in})|0\rangle . \end{equation*} In this expression we use that \(\langle (\vec{y},t_2);t|\) is the hermitean conjugate of \(|(\vec{y}, t_2); t \rangle \) and we evolve\(|(\vec{x}, t_1); t_1 \rangle \) to \(|(\vec{x}, t_1); t_2 \rangle \). In the Heisenberg picture the propagator reads \begin{equation*} G(\vec{y}, t_2; \vec{x}, t_1) = \langle 0 | a_H(\vec{y},t_2) a^\dagger _H(\vec{x},t_1)|0\rangle . \end{equation*} This follows from the identity \begin{equation*} U(t_2,t_1)= U(t_2,t_{in}) U(t_{in},t_1)=U(t_2,t_{in}) U^\dagger (t_{in},t_1) \end{equation*} and the definition of Heisenberg operators with reference point \(t_{in}\). The transition amplitude \(G\) is called the propagator or Green’s function. It is a central quantity in quantum field theory.

One particle wave function and Schrödinger equation. Before going on to derive the functional integral expression for the propagator we discuss next the Schrödinger equation for a one-particle state. This makes the connection to the standard formulation of quantum mechanics. Quantum mechanics obtains from quantum field theory by a restriction to states with a fixed particle number, typically a single particle or two particles. Since quantum field theory is quantum mechanics for many particles, it contains as a special case the quantum mechanical systems with a small fixed particle number. For a single particle we expect in our case the rather trivial quantum mechanics of a free particle, since we consider a translation invariant situation with a vanishing potential. In the functional integral formulation we could introduce a potential in the formulation of the action. For non-relativistic bosons one replaces the chemical potential by \(\mu -V(\vec{x})\), thus breaking translation symmetry.

We first extract the Schrödinger wave function in the position basis. Using our basis of localised one-particle states a general one-particle wave function at time \(t\) is a superposition \begin{equation*} |\psi _1(t)\rangle = \int _{\vec{x}} \varphi (\vec{x},t) | (\vec{x},t);t\rangle . \end{equation*} The position representation of the one-particle state or Schrödinger wave function is given by \(\varphi (\vec{x},t)\). As usual it can be extracted from \(|\psi (t)\rangle \) by \begin{equation*} \varphi (\vec{x},t) = \langle (\vec{x},t);t | \psi _1(t)\rangle \end{equation*} The proof is standard, using the orthogonality of basis functions \begin{equation*} \begin{split} \langle (\vec{x},t);t | \psi _1(t)\rangle &= \int _{\vec y} \langle (\vec{x},t);t| \varphi (\vec{y},t)|(\vec{y},t);t\rangle \\ &= \int _{\vec y} \varphi (\vec y,t) \langle (\vec{x},t);t|(\vec{y},t);t\rangle \\ &= \int _{\vec y} \varphi (\vec y,t) \delta (\vec{x}-\vec{y})\\ &= \varphi (\vec{x},t). \end{split} \end{equation*}

From the position representation we can switch to the momentum representation \(\phi (\vec{p},t)\) by a Fourier transform. For the momentum representation the evolution is trivial, \begin{equation*} i \partial{_t} \phi (\vec{p},t) =\left(\frac{\vec{p}^2}{ 2M}-\mu \right)\phi (\vec{p},t). \end{equation*} This follows by applying to \(|\psi _1 \rangle \) an infinitesimal evolution operator \begin{equation*} |\psi _1(t+d t) \rangle = U(t+ d t,t) |\psi _1(t) \rangle = -i \hat{H} d t |\psi _1(t) \rangle , \end{equation*} and noting that \(\hat{H}\) is diagonal in the momentum basis. The Schrödinger equation in position space obtains by a Fourier transform.

Huygens principle. You have learned before how to use a propagator for the evolution of wave functions, for example in electrodynamics. Our definition of the propagator plays exactly this role. We employ the time evolution of the position representation of the one particle wave function which can be found from the time evolution of \(|(\vec{x},t_1); t \rangle \), \begin{equation*} \begin{split} \varphi (\vec{y},t_2) &= \langle (\vec{y},t_2);t_2| \psi _1(t_2)\rangle \\ & =\langle (\vec{y},t_2);t_2| U(t_2, t_1) |\psi _1(t_1)\rangle \\ &= \int _{\vec{x}} \varphi (\vec{x},t_1)\langle (\vec{y},t_2);t_2|(\vec{x},t_1);t_2 \rangle \\ &= \int _{\vec{x}} G(\vec{y},t_2; \vec{x},t_1) \varphi (\vec{x}, t_1). \end{split} \end{equation*} The propagator \(G\) allows one to compute the one-particle wave function at \(t_2\) from an initial wave function at \(t_1\). This is Huygens’ principle for the propagation of waves.