Quantum field theory 1, lecture 11

Transition to field theory

Let us now go to quantum field theory. Instead of the positions \(x_1,\ldots, x_N\), the degrees of freedom are now the field variables \(\phi(\mathbf x)\) at some fixed time \(t\), for all possible spatial positons \(\mathbf x\). The spatial position \(\mathbf x\) now plays the role of the index \(n=1,\cdots,N\) and labels the different degrees of freedom (quantum fields).

Pure and mixed states

A pure state at some time \(t\) is now specified by a so-called Schrödinger functional \(\psi_t[\phi],\) and a mixed state in a similar way by a density matrix functional \(\rho_t[\phi_+,\phi_-].\) The most general observable is also specified by a similar functional \(A[\phi_1,\phi_2],\) and an expectation value is given by \[\langle A \rangle = \int D\phi_+ D\phi_- \,\rho_t[\phi_+,\phi_-] A[\phi_-,\phi_+].\] The functional integrals \(\int D\phi_+\) and \(\int D\phi_{-}\) are here over fields at constant time \(t\) but for all spatial positions \(\mathbf x\). The definition is as we have studied it for statistical field theory.

Conjugate momentum field

In this “position space” representation of a field theoretic state, the conjugate momentum field corresponds to an operator, \[\pi(t,\mathbf x)[\phi_1,\phi_2] = \left[ - i \frac{\delta }{\delta \phi_1(\mathbf x)} \right] \delta[\phi_1 - \phi_2],\] with the “functional Dirac delta” \(\delta[\phi_1-\phi_2]\) defined such that for some functional \(f[\phi]\) \[\int D\phi_1 f[\phi_1] \delta[\phi_1-\phi_2] = f[\phi_2].\] With this one obtains the expection value \[\langle \pi(t, \mathbf x) \rangle = \int D\phi \, \left[ - i \frac{\delta }{\delta \phi_+(\mathbf x)} \rho_t[\phi_+, \phi_-] \right]_{\phi_+=\phi_-=\phi}.\] In this way one can now calculate all kind of observables at some given time \(t\).

Ground state and excited states

Field theory of a single mode

To get an intuition, let us consider the simple case of a field theory in \(d=1+0\) dimensions. This describes a single field mode and has applications for example to cavity-quantum electrodynamics. The Lagrangian for a real variable \(\phi\) is \[L = \frac{1}{2}(\partial_t\phi)^2- \frac{1}{2}m^2\phi^2,\] and it is equivalent to the harmonic oscillator. This means that we can easily take over some results known from quantum mechanics.

Ground state

By recalling results from quantum mechanics for the harmonic oscillator, the Schrödinger functional for the ground state can be immediately specified as a Gaussian, \[\Psi_0[\phi] = c\exp\left(-\frac{m}{2}\phi^2 \right)\] with a complex constant \(c\). Accordingly, the density functional in that state \[\rho_0[\phi_+,\phi_-] = \frac{1}{Z} \exp\left(-\frac{m}{2}(\phi_+^2+\phi_-^2) \right).\] One can directly see that this is a pure state because it factorizes into a ket and a bra contribution.

Excited states

Excited states with \(n\) particles or quanta are of the form \[\rho_n[\phi_+,\phi_-] = \frac{1}{Z_n} H_n(\sqrt{m}\phi_+) H_n(\sqrt{m}\phi_-) \exp\left(-\frac{m}{2}(\phi_+^2+\phi_-^2) \right)\] where \(H_n(x)\) are the Hermite polynomials \[H_0(x) = 1,\qquad H_1(x) = 2x,\qquad H_2(x) = 4x^2-2, \cdots\] These are still pure states. The corresponding Schrödinger functional would be \[\Psi_n[\phi] = \frac{1}{\sqrt{2^n n!}} H_n(\sqrt{m}\phi) c \exp\left( -\frac{m}{2}\phi^2 \right).\] Under time evolution, the Schrödinger functionals above would pick up a factor \(e^{-im(n+1/2)t}\) which cancels, however, in the density functional.

Coherent states

Another interesting class of states are coherent states. In quantum mechanics they are described by \[|\alpha\rangle = e^{-|\alpha|^2/2} \sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}}|n\rangle = e^{-|\alpha|^2/2} e^{\alpha a^\dagger} |0\rangle,\] with complex parameter \(\alpha\). Here they lead to the density matrix functional \[\rho_\alpha[\phi_+,\phi_-] = \frac{1}{Z} \exp\left( -\frac{1}{2}m \left[ \left(\phi_+- \sqrt{ \frac{2}{m}}\text{Re}(\alpha)\right)^2 + \left(\phi_- - \sqrt{ \frac{2}{m}}\text{Re}(\alpha)\right)^2 \right]\right).\] Again these are pure states. Under time eolution, one must replace \(\alpha\to \alpha(t_0) e^{-im(t-t_0)}\) and one finds that \(\text{Re}(\alpha(t))\) describes the oscillatory behaviour of classical solutions to the equations of motion. The density matrix \(\rho_{\alpha(t)}\) describes Gaussian fluctuations around this mean value.

Themal states for a single mode

Finally, let us consider a thermal state. In the quantum mechanical formalism, it is described as \[\rho = (1-b) \sum_{n=0}^\infty b^n |n\rangle\langle n|,\] where \(b=e^{-m/T}\) is the Boltzmann weight. Here this leads to the density matrix functional \[\rho_T[\phi_+,\phi_-] = \frac{1}{Z}(1-b)\sum_{n=0}^\infty \frac{1}{n!} \left( \frac{b}{2}\right)^n H_n(\sqrt{m}\phi_+) H_n(\sqrt{m}\phi_-) e^{-m(\phi_+^2+\phi_-^2)/2}.\] Here, one can use a property of the Hermite polynomials (Mehler’s formula) \[\sum_{n=0}^\infty \frac{1}{n!} H_n(x) H_n(y) \left(\frac{b}{2} \right)^n = \frac{1}{\sqrt{1-b^2}} \exp\left[ \frac{2b}{1+b}xy - \frac{b^2}{1-b^2}(x-y)^2 \right].\] We thus find \[\begin{split} \rho_T[\phi_+,\phi_-] &= \frac{1}{Z} \exp\left[ - \frac{1}{2}m(\phi_+^2+\phi_-^2)- \frac{b^2}{1-b^2}m(\phi_+-\phi_-)^2+\frac{2b}{1+b}m\phi_+\phi_- \right]\\ &= \frac{1}{Z} \exp \left[ - \frac{1}{2}m \left( 1 + \frac{2b^2}{1-b^2}\right)(\phi_+^2+\phi_-^2) +\frac{2b}{1-b^2}m\phi_+\phi_- \right]. \end{split}\label{eq:01DensityMatrixThermal}\] This does not factor into a ket and a bra part for \(b>0\). It is therefore not a pure state as expected.

Gaussian versus non-Gaussian states

Let us summarize this discussion by remarking that the vacuum or ground state, the coherent states, as well as the thermal states all have density matrices \(\rho[\phi_+,\phi_-]\) of Gaussian form. This is not the case for single or multiple particle excited states, though. For free quantum field theories, one can also expect Gaussian states in many circumstances. However, already with non-vanishing interaction this ceases to be the case.

Higher dimensional Gaussian states

Let us now generalize the situation somewhat and consider a set of fields \(\phi_n\). The index \(n\) is here taken to be discrete and can run over a finite set of modes for example. However, it could be running over an infinite set. One may even consider \(n\) to be an abstract index that combines several indices such as momentum, flavor or spin.

We assume the Schrödinger functional to be of the form \[\Psi[\phi] = c \exp\left[ -\frac{1}{2}(\phi-\bar \phi)_m h_{mn} (\phi-\bar \phi)_n + ij_n\phi_n \right],\] with a symmetric and real matrix \(h_{mn}=h_{nm}\). The density functional for this pure state is accordingly \[\begin{split} \rho[\phi_+,\phi_-] &= \frac{1}{Z}\exp\bigg[ -\frac{1}{2}(\phi_+-\bar \phi)_m h_{mn} (\phi_+-\bar \phi)_n\\ &\qquad\qquad\quad -\frac{1}{2}(\phi_--\bar \phi)_m h_{mn} (\phi_--\bar \phi)_n + ij_n(\phi_+-\phi_-)_n \bigg]. \end{split}\nonumber\]

Characterization through one- and two-point functions

Let us characterize this state by its expectation values and correlation functions. Besides the field \(\phi_n\), another observable is its conjugate momentum field \(\pi_n\). In the position space representation, we are working in here, it is represented by a derivative \[\pi_n = -i \frac{\delta}{\delta\phi_n}.\] This operator acts on the Schrödinger functional or density operator. The canonical commutation relations \[[\phi_m,\pi_n] = i\delta_{mn},\qquad [\phi_m,\phi_n]=[\pi_m,\pi_n]=0,\] are automatically fulfilled.

The field expectation value is given by \[\langle \phi_m\rangle = \frac{1}{Z}\int D\phi~\phi_m\rho[\phi,\phi] = \bar\phi_m.\] In a similar way, the expectation value for the conjugate momentum can be obtained, \[\langle \pi_m\rangle = \frac{1}{Z}\int D\phi\left( -i \frac{\delta}{\delta\phi_{+m}}\rho[\phi_+,\phi_-] \right)_{\phi_+=\phi_-=\phi} = j_m.\] An exercise in Gaussian integration yields the connected correlation functions \[\begin{split} & \langle\phi_m\phi_n\rangle_c = \langle\phi_m\phi_n\rangle -\langle\phi_m\rangle\langle\phi_n\rangle = \frac{1}{2}(h^{-1})_{mn}, \quad\quad\quad \langle \pi_m,\pi_n\rangle_c = \frac{1}{2}h_{mn}, \\ & \langle\phi_m\pi_n+\pi_n\phi_m\rangle_c = 0, \quad\quad\quad \langle\phi_m\pi_n-\pi_n\phi_m\rangle_c = [\phi_m,\pi_n] = i\delta_{mn}. \end{split}\nonumber\] If the matrix \(h_{mn}\) is diagonal \(h_{mn}=\tilde h_m\delta_{mn}\) (no sum convention), the different field modes are independent, otherwise they are correlated.

Uncertainty relation

Imagine now that \(h_{mn}\) is diagonal. One then has \[\langle\phi_m^2\rangle\langle\pi_m^2\rangle = \frac{1}{4}.\] This is in fact the statement that Heisenberg’s uncertainty bound is satisfied and saturated. Note that for a single mode in the ground state, we have \[\langle \phi^2 \rangle = \frac{1}{2m}, \qquad \langle \pi^2 \rangle = \frac{m}{2}.\] The energy \(E=m\) here sets the quantum uncertainty. In quantum optics, it is possible, however, to prepare so-called squeezed states with \[\langle \phi^2 \rangle = \frac{1}{2h},\qquad \langle \pi^2 \rangle = \frac{h}{2},\] where \(h>m\) or \(h<m\). These are still pure states and they are still Gaussian states. They also still satisfy the Heisenberg bound but, for \(n>m\), have a reduced uncertainty of the field at the cost of an increased uncertainty of the conjugate momentum. For \(n<m\), the uncertainty of \(\pi\) is reduced while the one of \(\phi\) is increased.

For diagonal \(h_{mn}\), the different modes \(\phi_m\) are fully independent and the density matrix \(\rho[\phi_+,\phi_-]\) decomposes into a product of independent factors. This indicates that these degrees of freedom are not entangled. The situation is different in the presence of off-diagonal terms in \(h_{mn}\). In that case, there are non-vanishing correlations between fields and between conjugate momenta - but there is also quantum entanglement.

When quantum field theory is developed from a version of the theory with lattice regularization one finds that the field degrees of freedom in position space are strongly entangled by the spatial derivative term. For a quantum field theory entanglement is in this sense ubiquitous.

Entanglement

Two-mode squeezed state

As the simplest example for an entangled Gaussian state consider the two-mode squeezed state with Schrödinger functional \[\Psi_r[\phi_1,\phi_2] = c\exp\left[ - \frac{e^{2r}}{4} m (\phi_1-\phi_2)^2- \frac{e^{-2r}}{4}m(\phi_1+\phi_2)^2 \right].\] For \(r=0\), this simply becomes the product state \[\Psi_0[\phi_1,\phi_2] = c\exp\left[ - \frac{1}{2}m(\phi_1^2+\phi_2^2) \right] = c\exp\left[ -\frac{1}{2}m\phi_1^2 \right] \exp\left[ -\frac{1}{2}m\phi_2^2 \right].\] For \(r>0\), such a product decomposition is not possible, however. Generalizations of such two-mode squeezed states describe entangled states from inflation in the early universe or the entanglement of Hawking radiation emerging from a black hole with radiation falling into the horizon (for free bosonic theories). The density matrix for the two-mode system in the squeezed state is \[\begin{split} \rho_{12}[\phi_{1+},\phi_{2+};\phi_{1-},\phi_{2-}] &= \frac{1}{Z} \exp\bigg[ - \frac{e^{2r}}{4} m (\phi_{1+}-\phi_{2+})^2- \frac{e^{-2r}}{4}m(\phi_{1+}+\phi_{2+})^2\\ &\qquad\qquad\quad - \frac{e^{2r}}{4} m (\phi_{1-}-\phi_{2-})^2- \frac{e^{-2r}}{4}m(\phi_{1-}+\phi_{2-})^2 \bigg]. \end{split}\nonumber\]

Reduced density matrix

It is instructive to calculate the reduced density matrix for the mode \(\phi_1\) by taking the partial trace of the density matrix. Quite generally, the reduced density matrix for a subsystem \(A\) of a larger system consisting of the parts \(A\) and \(B\) is given as the partial trace \[\rho_A = \text{Tr}_B\left\{ \rho_{AB} \right\}.\] If \(A\) and \(B\) are entangled and \(\rho_{AB}\) describes a pure state, the reduced density matrix is of a mixed state form. In contrast, for a pure product state \(\rho_{AB} = \rho_A\otimes \rho_B\), the reduced density matrix \(\rho_A\) is also pure. In the present case, taking the partial trace for the second mode corresponds to \[\label{eq:01reallyLong} \begin{split} \rho_1[\phi_{1+},\phi_{1-}] = & \int D \phi ~\rho_{12}[\phi_{1+},\phi;\phi_{1-},\phi]\\ = & \frac{1}{Z} \int D \phi~\exp\bigg[ - \frac{e^{2r}+e^{-2r}}{4}m(\phi_{1+}^2+\phi_{1-}^2) \\ & +2m\phi \left( \frac{e^{2r}-e^{-2r}}{4}m(\phi_{1+}+\phi_{1-}) \right) -m \phi^2 \frac{e^{2r}+e^{-2r}}{2} \bigg] \\ = & \frac{1}{Z} \exp\bigg[ -\frac{1}{2}m\cosh(2r)(\phi_{1+}^2+\phi_{1-}^2) + \frac{1}{4}m \cosh(2r)\tanh^2(2r) (\phi_{1+}+\phi_{1-})^2 \bigg]\\ & \times \int D\phi \exp\bigg[ -m \cosh(2r)\left(\phi- \frac{1}{2}\tanh(2r)(\phi_{1+}+\phi_{1-})\right)^2 \bigg] \\ = & \frac{1}{Z} \exp\bigg[ -\frac{1}{2}m\cosh(2r) \left(1-\frac{1}{2}\tanh^2(2r)\right) (\phi_{1+}^2+\phi_{1-}^2) \\ & + \frac{1}{2}m \cosh(2r)\tanh^2(2r) \phi_{1+}\phi_{1-} \bigg]. \end{split}\] In the last step, we have performed the Gaussian integral over \(\phi\) and dropped an irrelevant factor.

As expected, for \(r>0\), the density matrix \(\rho_1\) now is not of pure state form anymore. It does not factor into a ket and a bra because of the term \(\propto \phi_{1+}\phi_{1-}\) in the exponent.
Note the resemblance of \(\eqref{eq:01reallyLong}\) and \(\eqref{eq:01DensityMatrixThermal}\). This is an indication that entanglement can sometimes lead to a locally thermal looking state, albeit it is globally pure.