Quantum field theory 1, lecture 12

Dynamics

So far we have been concerned with the description of states in the field theory, which can be specified for example at a globally fixed constant time \(t=t_0\). Our next goal is to understand dynamics in time. Before going there it is worth to clarify an issue related to relativistic causality.

States on Cauchy surfaces

It seems a bit strange that time is singled out for the description of states, because, according to special relativity, observers that move with a velocity relative to each other have different notions of what equal time actually means. Indeed, states can be specified somewhat more generally on any Cauchy surface \(\Sigma\). This is a \((d-1)\) dimensional submanifold of \(d\)-dimensional spacetime, a so-called hypersurface, with a normal vector that points in a time-like direction everywhere. A hypersurface \(t=t_0\) with normal vector \(n^\mu=(1,0,0,0)\) is then just a special case.

In the more general case, the density matrix on the hypersurface \(\Sigma\) is specified as a double functional of fields \(\phi_+(x)\) and \(\phi_-(x)\) where the coordinates are now on the hypersurface, that is \(x\in \Sigma\), \(\rho = \rho_\Sigma[\phi_+,\phi_-]\). In this formulation, a generalization of time evolution would correspond to an evolution between neighbouring Cauchy surfaces, e. g. \(\Sigma_1 \to \Sigma_2 \to \cdots \to \Sigma_N\). Keeping this generalization in mind, we can still take evolution according to some globally defined time coordinate as a convenient special case in the following.

Unitary time evolution

Similar as in quantum mechanics, the evolution in time, or between Cauchy surfaces, is realized by unitary evolution operators. For \(N\)-body quantum mechanics, this would be an operator of the type \[\begin{split} U_{t_2\leftarrow t_1}(\mathbf x_1,\cdots,\mathbf x_N; \mathbf y_1,\cdots,\mathbf y_N), \end{split}\] such that \[\begin{split} \psi_{t_2}(\mathbf x_1,\cdots,\mathbf x_N) &= \int_{\mathbf y_1,\ldots,\mathbf y_N} U_{t_2\leftarrow t_1}(\mathbf x_1,\cdots,\mathbf x_N; \mathbf y_1,\cdots,\mathbf y_N) \psi_{t_1}(\mathbf y_1,\cdots,\mathbf y_N). \end{split}\] The density matrix also needs the hermitian conjugate operator \[\begin{split} U^\dagger_{t_1\to t_2}(\mathbf x_1,\cdots,\mathbf x_N;\mathbf y_1,\cdots,\mathbf y_N) \end{split}\] so that the density matrix evolves as \[\begin{split} \rho_{t_2}(\mathbf x_1,\cdots,\mathbf x_N;\mathbf y_1,\cdots,\mathbf y_N) &= \int_{\mathbf u_1,\cdots,\mathbf u_N}\int_{\mathbf v_1,\cdots,\mathbf v_N} U_{t_2\leftarrow t_1}(\mathbf x_1,\cdots,\mathbf x_N; \mathbf u_1,\cdots,\mathbf u_N) \cdot \\ &\cdot \rho_{t_1} (\mathbf u_1,\cdots,\mathbf u_N; \mathbf v_1,\cdots,\mathbf v_N) U^\dagger_{t_1\to t_2}(\mathbf v_1,\cdots,\mathbf v_N;\mathbf y_1,\cdots,\mathbf y_N) \end{split}\] In a quantum field theory, one can specify in a similar way the unitary operator for evolution from one hypersurface to the next, e.g. \(\Sigma_1\to\Sigma_2\), or \(t_1 \to t_2\) \[\begin{split} U_{t_2\leftarrow t_1}[\phi_2,\phi_1], \end{split}\] such that the density matrix functional evolves as \[\label{eq:densityEv} \begin{split} \rho_{t_2}[\phi_{2+},\phi_{2-}] =\int D\phi_{1+} \int D\phi_{1-} U_{t_2\leftarrow t_1}[\phi_{2+},\phi_{1+}] \rho_{t_1}[\phi_{1+},\phi_{1-}] U^\dagger_{t_1\to t_2}[\phi_{1-},\phi_{2-}]. \end{split}\] This evolution equation of the density matrix functional is a special case of the general evolution equation for the density matrix in quantum mechanics \[\rho_{t_2} = e^{-iH(t_2-t_1)}\rho_{t_1}e^{iH(t_2-t_1)}.\] The left operator evolves the “ket” forward in time, while the right operator evolves the “bra” forward.

Schwinger-Keldysh double time path

The evolution operator for the “bra” \(e^{iH(t_2-t_1)}\) or \[U^\dagger_{t_1\to t_2}[\phi_{1-},\phi_{2-}]\] can also be understood as an operator that evolves backward in time. This is the idea beyond the Schwinger-Keldysh double time path that can be used to describe the time evolution for quantum field theories in general out-of-equilibrium situations. This is needed for example in cosmology or to describe non-equilibrium situations in condensed matter contexts. Note in particular that \(\rho_{t_1}\) can in principle be any density matrix. Because both the “ket” and the “bra” part of the density matrix are specified initially or as incoming one speaks of an in-in formalism. The outgoing state is not specified and must be calculated.

Functional integral for time evolution

Let us now consider the time evolution operator \[U_{t_\text{f}\leftarrow t_0}[\phi_\text{f},\phi_0].\] We are free to insert here intermediate steps here and to write \[U_{t_\text{f}\leftarrow t_0}[\phi_\text{f},\phi_0] = \int D\phi_N \cdots\int D\phi_2 \int D\phi_1 \, U_{t_\text{f}\leftarrow t_N}[\phi_\text{f},\phi_N] \cdots U_{t_2\leftarrow t_1}[\phi_2,\phi_1] U_{t_1\leftarrow t_0}[\phi_1,\phi_0] .\] By inserting many of these intermediate steps we can reduce everything to evolution operators over infinitesimal time steps \(t_{j+1}=t_j+\varepsilon\). For these one can write \[U_\varepsilon = e^{-i \varepsilon H} \approx \mathbb{1} - i \varepsilon H,\] with terms of quadratic order and higher vanishing in the limit \(\varepsilon\to 0\). The part \(\mathbb{1}\) corresponds here to a functional Dirac delta, which we can write as \[\delta[\phi_{j+1} - \phi_j] = \int D\pi_j \exp\left[ i \int d^{3} x \, \pi_j(\mathbf x) \left[\phi_{j+1}(\mathbf x) - \phi_j(\mathbf x)\right] \right]. \label{eq:01functionalDiracDelta}\] This is just the generalization of the familiar expression \[\delta(x-y) = \int \frac{d p}{2\pi} e^{i p(x-y)},\] to the functional formalism. The Hamiltonian is itself an operator that involves the fields \(\phi\) and its spatial derivatives, and the conjugate momentum operators \[-i \frac{\delta}{\delta \phi(\mathbf{x})}.\] When acting on an expression as in \(\eqref{eq:01functionalDiracDelta}\) this functional derivative operator gives just \(\pi_j(\mathbf x)\) under the integral.

Collecting terms we find \[\begin{split} U_{t_\text{f}\leftarrow t_0}[\phi_\text{f},\phi_0] = & \int D \pi_N \int D\phi_N \cdots \int D\pi_1 \int D\phi_1 \int D\pi_0 \\ & \exp\left[i \sum_{j=0}^N \varepsilon \left\{ \int d^3 x \left\{ \pi_j(\mathbf x)\frac{\phi_{j+1}(\mathbf x) - \phi_j(\mathbf x)}{\varepsilon} \right\} - H[\phi_j, \pi_j] \right\} \right]. \end{split}\nonumber\] We have re-exponentiated here the term invoving the Hamiltonian and were a bit sloppy with the question how to order terms in the Hamiltonian. This is justified by the limit \(\varepsilon\to 0\) we want to take next. We can replace with \(t=t_0+j \varepsilon\) the fields, \(\phi_j(\mathbf x)\to \phi(t, \mathbf x)\), conjugate momentum fields, \(\pi_j(\mathbf x)\to \pi(t, \mathbf x)\) and \[\frac{\phi_{j+1}(\mathbf x) - \phi_j(\mathbf x)}{\varepsilon} \to \frac{\partial}{\partial t} \phi(t, \mathbf x) = \dot \phi(t, \mathbf x).\] Moreover, the sum over \(j\) in the exponent becomes an integral along time.

Phase space functional integral

We find thus the functional integral expression \[U_{t_\text{f}\leftarrow t_0}[\phi_\text{f},\phi_0] = \int D\pi \int D\phi \exp\left[ i \int_{t_0}^{t_\text{f}} dt \int d^3 x \left\{ \pi(x) \dot \phi(x) - \mathscr{H}\right\} \right],\] where \(\mathscr{H}\) is the Hamiltonian density. The functional integral includes now one integral over the field at each point in time and space between the initial and final time or Cauchy hypersurface. At the boundaries of the time interval (or on the bounding Cauchy surfaces) one must keep \[\phi(t_0, \mathbf{x}) = \phi_0(\mathbf{x}), \quad\quad\quad \phi(t_\text{f}, \mathbf{x}) = \phi_\text{f}(\mathbf{x}), \label{eq:01FunctIntBoundaryConditions}\] fixed. The integrals over the conjugate momentum fields are not constrained.

Langrangian functional integral

Finally, for theories where the Hamiltonian is quadratic in the conjugate momentum fields \(\pi(t, \mathbf x)\) one can easily perform the functional integral over \(\pi(t, \mathbf x)\). Besides an irrelevant overall constant, this implies to extremize with respect to \(\pi(x)\), which is effectively the Legendre transform to an integral over the Lagrangian in the exponential, \[U_{t_\text{f}\leftarrow t_0}[\phi_\text{f},\phi_0] = \int D\phi \exp\left[ i \int_{t_0}^{t_f} dt \int d^3 x \mathscr{L}(\phi, \partial_\mu \phi) \right], \label{eq:01EvolutionLagrangian}\] where \(\mathscr{L}(\phi, \partial_\mu \phi)\) is the Lagrangian density. Specifically, for the scalar field theory one has \[\mathscr{L} = \frac{1}{2} \dot \phi^2 - \frac{1}{2} \nabla\phi^2 - V(\phi) = - \frac{1}{2} g^{\mu\nu} \partial_\mu \phi \partial_\nu\phi - V(\phi).\] We use here the Minkowski metric with mainly plus convention, \(g_{\mu\nu} = \text{diag}(-,+,+,+)\), and the microscopic potential \[V(\phi) = \frac{1}{2} m^2 \phi^2 + \frac{\lambda}{4!} \phi^4.\] At the boundaries in time we still need to keep the fields fixed according to \(\eqref{eq:01FunctIntBoundaryConditions}\).