Quantum field theory 2, lecture 04

Analytic continuation or Wick rotation

At this point it is useful to recall the idea of Wick rotation or analytic continuation from Minkowski to Euclidean space. We rotatate the time coordinate into an imaginary direction by writing \[t = -i \tau, \quad\quad\quad dt = -i d\tau,\] and obtain for the action in Minkowski space \[\begin{split} i S[\phi] = & i \int dt \int d^{d-1}x \left\{ \frac{\partial}{\partial t}\phi^* \frac{\partial}{\partial t}\phi - \boldsymbol{\nabla}\phi^* \boldsymbol{\nabla}\phi - V(\rho) \right\} \\ = & - i^2 \int d\tau \int d^{d-1}x \left\{ - \frac{\partial}{\partial \tau}\phi^* \frac{\partial}{\partial \tau}\phi - \boldsymbol{\nabla}\phi^* \boldsymbol{\nabla}\phi - V(\rho) \right\} \\ = & - S_\text{E}[\phi]. \end{split}\] In the last step we employ the Euclidean action \[S_\text{E}[\phi] = \int d\tau\int d^{d-1}x \left\{ \frac{\partial}{\partial \tau}\phi^* \frac{\partial}{\partial \tau}\phi + \boldsymbol{\nabla}\phi^* \boldsymbol{\nabla}\phi + V(\rho) \right\}.\] This has the nice feature that it is manifestly real and positive, which also motivates the additional sign introduced in the relation between the analytically continued or Wick rotated action and \(S_\text{E}[\phi]\). Quantum field theory for a scalar field in Euclidean space is in fact a statistical field theory.

Let us also recall here that quantum field theory at non-zero temperature is emplying the Eucliden version of the theory. The time coordinate \(\tau\) is integrated in the Matsubara formalism from \(\tau=0\) to \(\tau=1/T\) and bosonic fields have periodic boundary conditions, \(\phi(0,\mathbf{x}) = \phi(1/T,\mathbf{x})\).

Functional integral representation in Euclidean space

For defineteness we also give the different functional integral expressions in Euclidean space. The Schwinger functional can be written as \[e^{W_\text{E}[J]} = \int d\phi\, e^{-S_\text{E}[\phi] + \int_x \{ J^* \phi + \phi^* J \}}.\] The quantum effective action is again defined as the Legendre transform, \[\Gamma_\text{E}[\Phi] =\sup_J \left( \int_x\{ J^* \Phi + \Phi^* J\} - W[J] \right),\] and the same steps as in Minkowski space lead to the background field identity \[\exp\left(-\Gamma_\text{E}[\Phi]\right) = \int D\phi^\prime \exp\left(-S_\text{E}[\Phi + \phi^\prime] + \int d^dx \left\{ \frac{\delta\Gamma[\Phi]}{\delta\Phi} \phi^\prime + \phi^{\prime*} \frac{\delta \Gamma[\Phi]}{\delta\phi^*} \right\}\right).\] The concept of the one-particle irreducible effective action can also be used for classical statistical field theories and the action incorporates then the effect of statistical fluctuations.

Classical approximation

In classical field theory, fields are not fluctuating. This corresponds to the functional integral being completely dominated by the expectation value. With Euclidean conventions, the classical field approximation is accordingly \[\Gamma_\text{E}[\Phi] = S_\text{E}[\Phi].\] In Minkowski space there is an additional minus sign, due to the historic convention that the Lagrangian in mechanics is kinetic energy minus potential energy, \(L=T-U\), while the potential enters with a positive sign in the quantum effective potential. Accordingly, the classical approximation is there \[\Gamma[\Phi] = - S[\Phi].\] In reallity, quantum and statstical fluctuations can have a significant influence on the quantum effective action and lead to substantial deviations from the classical form.

An exception from this general rule occurs for theories that are quadratic in the fields. For an action of the from (emplying abstract index notation) \[S[\Phi+\phi^\prime] = \frac{1}{2} P_{\alpha\beta} (\Phi^\alpha+\phi^{\prime\alpha}) (\Phi^\beta+\phi^{\prime\beta})\] one finds \[\Gamma[\Phi] = S[\Phi] + \text{const}.\] Indeed, using \[\frac{\delta \Gamma[\Phi]}{\delta\Phi^\alpha} = P_{\alpha\beta} \Phi^\beta,\] the functional integral representation becomes \[\begin{split} \exp\left(-\frac{1}{2}P_{\alpha\beta} \Phi^\alpha\Phi^\beta - \text{const}\right) = & \int D\phi^\prime \exp\left( - S[\Phi+\phi] + \frac{\delta\Gamma}{\delta\Phi^\alpha} \phi^{\prime\alpha} \right) \\ = & \int D \phi^\prime \exp\left(-\frac{1}{2}P_{\alpha\beta} \Phi^\alpha\Phi^\beta -\frac{1}{2}P_{\alpha\beta} \phi^{\prime\alpha}\phi^{\prime\beta}\right). \end{split}\] This is solved for \[e^{-\text{const}} = \int D \phi^\prime \exp\left( -\frac{1}{2}P_{\alpha\beta} \phi^{\prime\alpha}\phi^{\prime\beta} \right).\] Usually, this constant or field-independent part can be simply dropped. But sometimes, for example in the context of thermodynamics, it depends on external parameters such as temperature, chemical potential, or external gauge field, and must then be taken into account, as well.

Perturbation theory

Interactions correspond to terms in \(S[\phi]\) that are not quadratic in \(\phi\), such as cubic or quartic terms. If the couplings characterizing the interaction are small, one expects some kind of perturbation expansion in the small couplings, \[\Gamma[\Phi]=S[\Phi]+\text{perturbative corrections}.\] We will explore this approach in more detail below.

Quantum vertices

The effective action \(\Gamma\) contains new vertices that are not present in the classical or microscopic action \(S\). For example photon-photon interactions. Classical Maxwell theory has no photon-photon interactions; Maxwell equations are linear. But the quantum one-particle irreducible four point function for photons contains terms like

This implies a one-loop contribution to photon-photon scattering, of order \[\Gamma^{(4)}\sim \alpha^2\sim e^{4}.\] For very small very small momenta below \(m_e\) one finds \[\Gamma^{(4)}\sim \frac{q^4}{m_e^4}.\] The correction is small, but observable by precisions measurements. Recently light-by-light scattering has been observed by ATLAS experiment in heavy ion collisions at the Large Hadron Collider.

Another example is \(g-2\), as generated by one-particle irreducible diagrams of the type

The corresponding piece of the effective action is of the form \[\Gamma \sim \int_x \bar\psi(x) \left[\gamma^\mu,\gamma^\nu \right]F_{\mu\nu}(x) \psi(x),\] where \(F_{\mu\nu}=\partial_\mu A_\nu- \partial_\nu A_\mu\) is the electromagnetic field strength.

An interesting fact about quantum vertices, i. e. terms that are not present on the level of the microscopic action, but that arise purely from quantum fluctuations, is that the corresponding coeffients can actually be determined from quantum field theory! Ultimately this is related to how renormalization works.

Effective potential

The part if the effective action for scalar fields that involves no derivative is the effective potential \(U(\rho)\), \[\Gamma[\Phi]=\int_x U(\rho) +\ldots.\] The ellipses are here for terms with derivatives which vanish for homogeneous fields. For the \(\text{O}(N)\)-symmetric scalar model, the effective potential can only depend on the invariant combination \[\rho =\frac{1}{2} \phi_n\phi_n.\] The exact quantum field equation for homogeneous fields \(\phi_n(x)=\phi_n\) becomes \[\frac{\partial}{\partial \phi_n} U(\rho) = \phi_n \frac{\partial}{\partial \rho} U(\rho) =0.\] There is always a solution \(\phi_n=0\), but this may not be the absolute minimum of \(U\). For reasons of stability the solution should be at least a local minimum. For a minimum at \(\rho_0\ne0\) one has spontaneous symmetry breaking, see next section.

Omitting fluctuations effects one has \(U(\rho)=V(\rho)\). In this limit the effective potential equals the microscopic potential. Quantum fluctuation induce a map \(V(\rho)\to U(\rho)\). This can also depend on external parameters, such as temperature or chemical potentials.

Higgs mechanism and superconductivity

Once the effective action is computed, or a given form is assumed, many properties of the system follow from the field equations and the correlation functions. Often one knows only the symmetries of \(\Gamma[\Phi]\) or the generic form of the effective potential, and uses an expansion in the number of derivatives \(\partial_{\mu}\Phi_j(x)\). The derivative expansion is motivated by the interest in macroscopic wave lengths, for which smooth fields often (not always!) play a dominant role. The validity of the derivative expansion may depend on the appropriate choice of macroscopic fields. In general the macroscopic fields can be more complicated than simply \(\Phi_j(x)=\langle\phi_j(x)\rangle\). An example are antiferromagnets.

Symmetry of the quantum effective action

Consider again a complex scalar field \(\phi(x)\), with \(S[\phi]\) invariant under global \(\text{U}(1)\) transformations, \(\phi(x)\to e^{i\alpha}\phi(x)\). Let us take a microscopic action which is \(\text{U}(1)\) invariant, \[S[\phi]= - \int_x \left\{g^{\mu\nu}\partial_{\mu}\phi^*\partial_{\nu}\phi+ V(\phi^*\phi) \right\} .\] If the functional measure \(\int D \phi\) is invariant under \(\text{U}(1)\) transformations, it follows that the quantum effective action \(\Gamma[\Phi]\) is also invariant under \(\text{U}(1)\) transformations, where \(\Phi(x)\to e^{i\alpha}\Phi(x)\).

To see this, note that the Schwinger functional \(W[J]\), defined through \[e^{iW[J]}=\int D\phi \, e^{iS[\phi]+i \int_x \{J^*\phi+\phi^*J\}} ,\] is invariant under the global \(\text{U}(1)\) transformations \(J(x)\to e^{i \alpha }J(x)\). Accordingly, the field expectation value \[\Phi(x)= \frac{1}{\sqrt{g}(x)}\frac{\delta W[J]}{\delta J^*(x)}\] transforms as \(\Phi(x)\to e^{i\alpha}\Phi(x)\), as one expects for \(\Phi(x)=\langle \phi(x)\rangle\). As a consequence, one finds that \(\int_x \{J^*\Phi+\Phi^*J\}\) is invariant. This establishes the invariance of \(\Gamma[\Phi]=\int_x\{J^*\Phi+\Phi^*J\} - W[J]\).

This generalizes to all symmetry transformations: If \(S[\phi]\) is invariant under some symmetry transformation \(\phi\to \mathsf{g} \phi\), and the functional measure \(\int D \phi\) is invariant under the transformation as well, \(\int D\phi = \int D\mathsf{g}\phi\), it follows that \(\Gamma[\Phi]\) is invariant under \(\Phi \to \mathsf{g}\Phi\). The effective action has the same symmetries as the classical action. This holds unless there is an “anomaly" in the functional measure.

Derivative expansion and Landau theories

In condensed matter physics, the precise microscopic physics is often not known, and the transition from microphysics to macrophysics (the computation of the quantum effective action) is very difficult. In addition, very different microphysical systems often give similar macroscopic phenomena. This is called universality and plays a crucial role for classical statistical field theories that describe condensed matter systems in the vicinity of second order phase transitions.

A useful approach is a guess for the quantum effective action \(\Gamma[\Phi]\). From comparison with experiment and general considerations one makes an assumption on what are the relevant macroscopic degrees of freedom \(\phi_j(x)\), without necessarily knowing the microscopic origin. Examples are spin waves for antiferomagnetism, or a complex scalar field \(\varphi(x)\) for superconductivity. The miscroscopic degrees of freedom are electrons, and the macroscopic field may represent Cooper pairs or similar composite objects. A second central ingredient is an assumption about the symmetries of the quantum effective action. Third, one employs a derivative expansion, typically up to two derivatives \(\partial_{\mu}\phi(x)\). This restricts the effective action already severely. For the example of the scalar \(\text{O}(N)\) model one remains with three functions of the invariant combination \[\rho=\frac{1}{2}\sum_{n=1}^N \Phi_n\Phi_n,\] appearing in the action \[\Gamma[\Phi]=\int_x \left\{ U(\rho) + \frac{1}{2}Z(\rho) g^{\mu\nu} \partial_{\mu}\Phi_n \partial_\nu\Phi_n +\frac{1}{4} Y(\rho) g^{\mu\nu}\partial_\mu \rho \partial_\nu \rho \right\}.\] This approach can be used in very similar form for relativistic fields in Minkoski space, for condensed matter systems in real time, where the velocity of light is replaced by a smaller velocity for some collective excitations, or for statistical field theories in Euclidean spaces. Even an extension to non-relativistic space-times is possible.

Making further assumptions, as a polynomial expansion of \(U(\rho)\) around its minimum and constant \(Z\) and \(Y\), one ends with a few parameters. These parameters may be fixed by comparison with experiment. For thermodynamics, they can depend on \(T\) and \(\mu\). This approach is very successful to gain physical insight without knowledge of the microphycis. The aim of quantum or statistical field theory is to do better by computing the free couplings or relations between them.

For many questions, the most important quantity is the effective potential. For the scalar \(O(N)\) model one may write \[U(\rho)= m^2 \rho + \frac{\lambda}{2}\rho^2+\ldots\] In lowest order one has two couplings \(m^2\) and \(\lambda\). One further assumes constant \(Z\), and the fields can be normalized such that \(Z(\rho)=1\). The coupling \(Y\) can often be dropped, \(Y(\rho)=0\). We will concentrate mainly on \(N=2\). The symmetry \(\text{U}(1) = \text{O}(2)\) is an abelian symmetry. We employ again the complex field \(\Phi(x)\), with \(\rho=\Phi^*\Phi\). Our “Landau theory” is \[\Gamma[\Phi]=\int_x\left\{ g^{\mu\nu} \partial_{\mu}\Phi^* \partial_{\nu}\Phi + m^2\Phi^*\Phi+ \frac{\lambda}{2}(\Phi^*\Phi)^2\right\},\] where the quartic coupling \(\lambda\) determines the strength of the interaction.

Spontaneous symmetry breaking

Spontaneous symmetry breaking is a key concept in condensed matter and particle physics. It extends to other branches of science as well. The basic ingredient is an effective action that has a given symmetry, while the solution of the field equation breaks this symmetry. The most important example is an effective potential with a minimum at \(\Phi \neq 0\). In a Euclidean setting the stable solution of the field equations is the “ground state”. It typically corresponds to a minimum of \(\Gamma[\Phi]\). We include here the possibility of a local minimum, which would corresponds to a metastable state. A positive kinetic term is minimized by a homogeneous field \(\Phi(x)=\Phi_0\), where \(\partial_\mu\Phi(x)=0\). The minimum of \(\Gamma[\Phi]\) corresponds then to a minimum of the effective potential \(U(\rho)\).

There are two general possibilities for \(U(\rho)\). The first is that the minimum is at \(\rho_0=0\) which implies that \(\Phi_0=0\) is invariant under the symmetry \(\text{U}(1)\). This is called the “symmetric phase”.

The second possibility is that the minimum of \(U(\rho)\) is at \(\rho_0>0\), such that the field expectation value \(\Phi_0\neq 0\) is not invariant under the symmetry group \(\text{U}(1)\). This is a “phase with spontaneous symmetry breaking”.

A potential with this shape is often called “mexican hat potential”, since it is rotation symmetric around the point \(\Phi=0\). The phase of \(\Phi_0\) is not determined! Every phase of \(\Phi_0\) is equivalent, but the ground state must pick up a fixed direction! Another example for spontaneous symmetry breaking is a rotationally symmetric stick under the influence of gravity. The rotation symmetric state of a vertical stick is unstable, and the ground state of a horizontal stick lying on the floor breaks rotation symmetry spontaneously. Other examples are magnets for which the expectation value of the spin in a Weiss domain singles out some direction.