Quantum field theory 2, lecture 01

Generating functionals

Partition function

We start with the partition function for a complex scalar field in the presence of a source field \(J\), \[Z[J] = \int D\phi \exp\left[iS[\phi] + \int d^d x \left\{ J^*(x) \phi(x) + \phi^*(x) J(x) \right\} \right]\] We work with a complex field \(\phi\) which we could also express in terms of real and imaginary parts as \[\phi(x) = \frac{1}{\sqrt{2}} \left[ \phi_1(x) + i \phi_2(x) \right],\] where \(\phi_1(x)\in \mathbb{R}\) with \(n=1,2\) are now real fields. Similarly for the source, \[J(x) = \frac{1}{\sqrt{2}} \left[ J_1(x) + i J_2(x) \right].\] With this, the source terms can also be written as \[J^*(x) \phi(x) + \phi^*(x) J(x) = J_1(x) \phi_1(x) + J_2(x) \phi_2(x).\] The theory we consider is a special case of an O\((N)\) model with \(N=2\).

Functional integral

The functional integral is an integral over \(\phi_1(x)\) and \(\phi_2(x)\) at every spacetime point, \[\begin{split} \int D\phi & = \int D\phi_1 \int D\phi_2 \\ & = \lim_{N\to\infty} \int_{-\infty}^\infty d\phi_1(x_1) \int_{-\infty}^\infty d\phi_2(x_1) \cdots \int_{-\infty}^\infty d\phi_1(x_N) \int_{-\infty}^\infty d\phi_2(x_N), \end{split}\] where \(x_1, \ldots, x_N\) corresponds to points on some lattice in \(d=1+3\) dimensional space-time, and \(N\to \infty\) corresponds to the continuum limit.

Transition amplitude

At the initial time \(t=t_\text{in}\) and at the final time \(t=t_\text{f}\) we can keep the field fixed, i. e. \[\begin{split} \phi(t_\text{in}, \mathbf{x}) & = \phi_\text{in}(\mathbf x), \\ \phi(t_\text{f}, \mathbf{x}) & = \phi_\text{f}(\mathbf x), \end{split}\] and, up to an overall factor, \(Z\) is then the quantum mechanical transition amplitude in the presence of sources, \[Z = U_{t_\text{f}\leftarrow t_\text{in}}[\phi_\text{f}, \phi_\text{in}].\] For \(t_\text{in}\to -\infty\), \(t_\text{f}\to \infty\), and using the \(i\epsilon\)-prescription, one has for \(J=0\) formally a vacuum-to-vacuum transition amplitude.

One can also use transition amplitudes such as \(U_{t_\text{f}\leftarrow t_\text{in}}\) to construct time evolution of density operators for non-equilibrium dynamics (Schwinger-Keldysh formalism).

Action

We now specify how the action \(S[\phi]\) typically looks like. Iy is supposed to be invariant under global \(\text{U}(1)\cong \text{O}(2)\) transformations, \[\phi(x) \to e^{i\alpha} \phi(x),\] and it should be local, i. e. an integral over a Lagrange density. In Minkowski space, with mainly plus conventions for the metric, \(\eta_{\mu\nu}=\text{diag}(-1,+1,+1,+1)\), it has the form \[S[\phi] = \int_{t_\text{in}}^{t_\text{f}} dt \int d^3 x \left\{ -\eta^{\mu\nu} \partial_\mu \phi^* \partial_\nu \phi - V(\rho) \right\},\] where \(\rho = \phi^*\phi = \frac{1}{2}[\phi_1^2+\phi_2^2]\) is invariant under \(U(1)\) transformations and the kinetic term can also be written as \[-\eta^{\mu\nu} \partial_\mu \phi^* \partial_\nu \phi = (\partial_t \phi^*)(\partial_t \phi) - \boldsymbol{\nabla}\phi^* \boldsymbol{\nabla}\phi.\]

Microscopic potential

The microscopic potential \(V(\rho)\) is usually expanded in a Taylor series around its minimum. This can either be at \(\rho=0\), in which case the expansion reads \[V(\rho) = m^2 \rho + \frac{1}{2}\lambda \rho^2+\ldots,\] where \(m\) is now the microscopic (or bare, or un-renormalized) mass of the scalar particles, and \(\lambda\) is similarly a microscopic (or bare or un-renormalized) interaction strength. We have assumed here for simiplicity that \(V(\rho)\) vanishes at its minimum, more generally a non-zero value would correspond to a cosmological constant.

The minimum can also be at a positive value \(\rho_0>0\), and the expansion reads then \[V(\rho) = \frac{1}{2}(\rho-\rho_0)^2 + \frac{1}{3!} (\rho-\rho_0)^3+\ldots.\] There is now no linear term. Besides \(\lambda\), the microscopic potential is then parametrized by the value of \(\rho_0\), and at higher order by \(\gamma\) etc.

Gauge fields

It is interesting and useful to extend the global U\((1)\) symmetry to a local symmetry by introducing an (external) gauge field \(A_\mu(x)\). We replace \[\begin{split} \partial_\mu \phi(x) & \to D_\mu \phi(x) = [\partial_\mu - i e A_\mu(x)] \phi(x), \\ \partial_\mu \phi^*(x) & \to D_\mu \phi^*(x) = [\partial_\mu + i e A_\mu(x)] \phi^*(x), \end{split}\] and have now an invariance under the local or gauge transformations \[\begin{split} \phi(x) & \to e^{i\alpha(x)}\phi(x),\\ \phi^*(x) & \to e^{-i\alpha(x)}\phi^*(x),\\ A_\mu(x) & \to A_\mu(x) + \frac{1}{e} \partial_\mu \alpha(x), \end{split}\] such that \(D_\mu\) transforms as a covariant derivative, \[D_\mu\phi(x) \to e^{i\alpha(x)} D_\mu\phi(x).\] One can understand \(e\) to be the electromagnetic charge of the scalar particles described by the field \(\phi(x)\). The action is now a functional of \(\phi\) and \(A_\mu\), and similarly the partition function \[Z[J,A] = \int D\phi \, e^{iS[\phi, A] + i \int_x \left\{ J^* \phi + \phi^* J \right\}}.\]

At present \(A_\mu(x)\) is just an external gauge field, but we could extend the theory such that it becomes a dynamical field which is also included in the functional integral like in quantum electrodynamics (QED). This is the scalar QED, because the matter fields are scalar particles and not Dirac fermions as in standard or spionor QED.

Electromagnetic current

From the variation of the action with respect to \(A_\mu(x)\) we can obtain the microscopic (or bare or un-renormalized) form of the electromagnetic or \(\text{U}(1)\) current, \[\delta S = \int d^x \delta A_\mu(x) J^\mu(x).\] Concretely, we find here \[J^\mu(x) = i e \left[ (D^\mu \phi^*(x)) \phi(x) - \phi^*(x) D^\mu\phi(x)\right].\] As a check, we eveluate this for \[\phi(x) = \frac{1}{\sqrt{2m}} \varphi(x) e^{-imt}, \quad\quad A_\mu(x) = 0,\] and obtain for a constant non-relativistic field \(\varphi(x)=\varphi_0\) \[J^0 = e \varphi^*_0 \varphi_0,\] corresponding to the charge times the density, as expected.

Action in general coordinates

It is actually useful to generalize our current formalism in another direction, namely by allowing general (not necessarily cartesian) coordinates. The Minkowski space metric becomes then space- and time-dependent, \[\eta_{\mu\nu} \to g_{\mu\nu}(x),\] and we could even go to curved space where we have no cartesian coordinates. This would be needed in the setup of Einsteins theory of gravitation, general relativity. The action becomes \[S = \int d^d x \sqrt{g} \left\{ - g^{\mu\nu}(x) D_\mu \phi^*(x) D_\nu \phi(x) - U(\rho) \right\}.\] We use here the invariant volume element \(d^dx \sqrt{g}\), with \(g=-\text{det} (g_{\mu\nu})\).

General coordinate transformations

A nice feature of the action is that it is invariant under general coordinate transformations, or diffeomorphisms, \[x \to x^\prime(x),\] with \[\begin{split} \frac{\partial}{\partial x^\mu} & \to \frac{\partial x^{\prime \alpha}}{\partial x^\mu} \frac{\partial}{\partial x^{\prime \alpha}}, \\ g_{\mu\nu}(x) & \to g^\prime_{\mu\nu}(x^\prime) = g_{\alpha\beta}(x(x^\prime)) \frac{\partial x^\alpha}{\partial^{\prime\mu}} \frac{\partial x^\beta}{\partial x^{\prime\nu}},\\ \sqrt{g} & \to \sqrt{g^\prime} = \sqrt{g} \, \det\left(\frac{\partial x^\alpha}{\partial x^{\prime\mu}}\right). \end{split}\] Exercise: Show that the action \(S\) is indeed invariant under this transformation.

Energy-momentum tensor

Similar to the electromagnetic current we can obtain the energy-momentum tensor from variation of the action with respect to the metric, \[\delta S = \frac{1}{2} \int d^d x \sqrt{g} \, T^{\mu\nu}(x) \delta g_{\mu\nu}(x).\] From the concrete expression for \(S\) we find, using the relations \[\delta g^{\mu\nu} = - g^{\mu\rho} g^{\nu\sigma} \delta g_{\rho\sigma},\] and \[\delta \sqrt{g} = \frac{1}{2} \sqrt{g} g^{\mu\nu} \delta g_{\mu\nu},\] the microscopic (or bare or un-renormalized) energy-momentum tensor \[T^{\mu\nu} = 2 D^\mu \phi^* D^\nu \phi - g^{\mu\nu}[g^{\alpha\beta}D_\alpha\phi^* D_\beta \phi + U(\rho)].\]