Prof. Dr. Stefan Floerchinger

Researcher and professor for quantum field theory.

Partition function and functional derivatives

The partition function for the model \(\eqref{eq:ONModel}\) reads \[Z[J] = \int D\phi \; e^{-S[\phi] + \int d^d x \{J_n(\mathbf{x}) \phi_n(\mathbf{x})\}} \label{eq:partitionfunctionON}\] We have introduced here an external source term \(\int d^d x \{J_n(\mathbf{x}) \phi_n(\mathbf{x})\}\) which can be used to probe the theory in various ways. For example, one can take functional derivatives to calculate expectation values, \[\langle \phi_n(\mathbf{x}) \rangle = \frac{1}{Z[J]} \frac{\delta}{\delta J_n(\mathbf{x})} Z[J] {\Big |}_{J=0},\] and correlation functions, e. g. \[\langle \phi_n(\mathbf{x}) \phi_m(\mathbf{y}) \rangle = \frac{1}{Z[J]} \frac{\delta^2}{\delta J_n(\mathbf{x}) \delta J_m(\mathbf{y})} Z[J]{\Big |}_{J=0} = \frac{\int D\phi \; \phi_n(\mathbf{x})\, \phi_m(\mathbf{y})\; e^{-S[\phi]}}{\int D\phi \; e^{-S[\phi]}}.\]

Classical field equation

In the the functional integral the contribution of field configurations \(\phi(\mathbf{x})\) is suppressed if the corresponding action \(S[\phi]\) is large. In the partition function \(\eqref{eq:partitionfunctionON}\), large contributions come mainly from the region around the minima of \(S[\phi]-\int_x J_n \phi_n\), which are determined by the equation \[\frac{\delta}{\delta \phi(\mathbf{x})} \left( S[\phi] - \int d^d x \{J_n(\mathbf{x}) \phi_n(\mathbf{x})\} \right) = \frac{\delta S[\phi]}{\delta \phi_n(\mathbf{x})} - J_n(\mathbf{x}) = 0.\] This equation is the field equation or equation of motion of a classical field theory. For the model \(\eqref{eq:ONModel}\) one has concretely \[\frac{\delta S[\phi]}{\delta \phi_n(\mathbf{x})} = - \partial_j\partial_j \phi_n(\mathbf{x}) + m^2 \phi_n(\mathbf{x}) + \frac{1}{2}\lambda \phi_n(\mathbf{x}) \phi_k(\mathbf{x}) \phi_k(\mathbf{x}) = J_n(\mathbf{x}).\] Note that this field equation is from a mathematical point of view a second order, semi-linear, partial differential equation. It contains non-linear terms in the fields \(\phi_n\), but the term involving derivatives is linear; therefore semi-linear. The equation involves the Euclidean Laplace operator \(\Delta = \partial_j \partial_j\) and is therefore of elliptic type (as opposed to hyperbolic or parabolic). This field equation is the correspondence of Maxwells equations in electrodynamics for our scalar theory. The source \(J\) corresponds to the electromagnetic current in Maxwell’s equations.

The \(O(N)\) symmetric potential

The model in \(\eqref{eq:ONModel}\) can be generalized somewhat to the action \[S[\phi] =\int d^d x \left\{ \frac{1}{2} \partial_j \phi_n \partial_j \phi_n + V(\rho) \right\}, \label{eq:ONModel2}\] where \(\rho = \frac{1}{2}\phi_n \phi_n\) is an \(O(N)\) symmetric combination of fields and \(V(\rho)\) is the microscopic \(O(N)\) symmetric potential. The previous case \(\eqref{eq:ONModel}\) can be recovered for \(V(\rho) = m^2 \rho + \frac{1}{2}\lambda \rho^2\).

More general, \(V(\rho)\) might be some function with a minimum at \(\rho_0\) and a Taylor expansion around it, \[V(\rho) = m^2 (\rho-\rho_0) + \frac{1}{2} \lambda (\rho-\rho_0)^2 + \frac{1}{3!} \gamma (\rho-\rho_0)^3 + \ldots\] If the minimum is positive, \(\rho_0>0\), the linear term vanishes of course, and one takes \(m^2=0\). In contrast, if the minimum is at \(\rho_0=0\) one has in general \(m^2>0\). In practice, one uses either \(\rho_0\) or \(m^2\) for a parametrization of \(V(\rho)\). It costs a certain amount of energy for the field to move away from the minimum. In particular, for large \(\lambda\) such configurations are suppressed.

Homogeneous solutions

It is instructive to discuss homogeneous solutions of the field equation, i.e. solutions that are independent of the space variable \(x\). For vanishing source \(J_n(\mathbf{x})=0\), and the model \(\eqref{eq:ONModel2}\) we need to solve \[\frac{\partial}{\partial \phi_n} V(\rho) = \phi_n \frac{\partial}{\partial \rho} V(\rho)= 0.\] This has always a solution \(\phi_n=0\) and for \(\rho_0=0\) and positive \(m^2\) this is indeed a minimum of the action \(S[\phi]\). For positive \(\rho_0\) the situation is more interesting, however. In that case, \(\phi_n=0\) is actually typically a maximum while the minimum is at \(\phi_k \phi_k = 2 \rho_0\), i. e. at a non-zero field value. One possibility is \(\phi_1=\sqrt{2\rho_0}\) with \(\phi_2=\ldots=\phi_n=0\), but there are of course many more. But such a solution breaks the \(O(N)\) symmetry! One says that the \(O(N)\) symmetry is here spontaneously broken on the microscopic level which technically means that the action \(S[\phi]\) is invariant, but the solution to the field equation (i. e. the minimum of \(S[\phi]\)) breaks the symmetry. It is an interesting and non-trivial question whether the symmetry breaking survives the effect of fluctuations. One has proper macroscopic spontaneous symmetry breaking if the field expectation value \(\langle \phi_n \rangle\) is non-vanishing and singles out a direction in field space. An example for spontaneous symmetry breaking is the magnetization field in a ferromagnet.

Constrained fields

It is also interesting to consider models where \(\rho=\rho_0\) is fixed. In fact, they arise naturally in the low energy limit of the models described above when the fields do not have enough energy to climb up the effective potential. Technically, this corresponds here to the limit \(\lambda\to \infty\) with fixed \(\rho_0\) and can be implemented as a constraint \[\phi_n(\mathbf{x}) \phi_n(\mathbf{x}) = 2 \rho_0. \label{eq:constraintNonLinearSigmaModel}\] Note that with this constraint, the field is now living on a manifold corresponding to the surface of an \(N\)-dimensional sphere, denoted by \(S_{N-1}\). One can parametrize the field as (the naming conventions are historic, one should not confuse the fields \(\pi_j\) with conjugate momentum fields) \[\phi_1 = \sigma, \quad \phi_2=\pi_1, \quad \ldots \quad \phi_N=\pi_{N-1},\] where only the fields \(\pi_n\) are independent while \(\sigma\) is related to them via the non-linear constraint \[\sigma = \sqrt{2\rho_0 - \vec \pi^2}.\]

Linearly and non-linearly realized symmetries

The symmetry group \(O(N)\) falls now into two parts. The first consists of transformations \(O(N-1)\) which only act on the fields \(\pi_n\) but do not change the field \(\sigma\). Such transformations are realized in the standard, linear way \[\pi_n \to O^{(N-1)}_{nm} \pi_m, \quad\quad\quad \sigma\to \sigma.\] In addition to this, there are transformations in the complement part of the group (rotations that also involve the first component \(\sigma\)). They act infinitesimally on the independent fields like \[\delta \pi_n = \delta\alpha_{n} \sigma = \delta\alpha_{n} \sqrt{2\rho_0 - \vec \pi^2}, \quad\quad\quad \delta \sigma = - \delta\alpha_{n} \pi_n,\] where \(\delta\alpha_{n}\) are infinitesimal parameters (independent of the fields). Note that this is now a non-linearly realized symmetry in the internal space of fields. This explains also the name non-linear sigma model.

Action

Let us now write an action for the non-linear sigma model. Because of the constraint \(\eqref{eq:constraintNonLinearSigmaModel}\), the effective potential term in \(\eqref{eq:ONModel2}\) becomes irrelevant and only the kinetic term remains, \[S[\pi] = \int d^d x \left\{ \frac{1}{2} \partial_j \phi_n \partial_j \phi_n \right\} = \int d^dx \left\{ \frac{1}{2} G_{mn}(\vec \pi) \partial_j \pi_m \partial_j \pi_n\right\}.\] In the last equation we rewrote the action in terms of the independent fields \(\pi_n\) and introduced the metric in the field manifold \[G_{mn}(\vec \pi) = \delta_{mn} + \frac{\pi_m\pi_n}{2\rho_0 - \vec \pi^2}.\] The second term originates from \[\partial_j \sigma = \partial_j \sqrt{2\rho_0 - \vec \pi^2} = \frac{1}{\sqrt{2\rho_0 - \vec \pi^2}} \pi_m \partial_j \pi_m.\]

Functional integral

Note that also the functional integral is now more complicated. It must involve the determinant of the metric \(G_{mn}\) to be \(O(N)\) invariant. For a single space point \(x\) one has \[\int \prod_n d\phi_n\to \int \prod_n d\phi_n \, \delta(\phi_n\phi_n -2 \rho_0) = \text{const} \times \int \sqrt{\text{det}(G(\vec \pi))}\,\prod_n d\pi_n.\] Only in the presence of the determinant \(\text{det}(G(\vec \pi))\) the functional measure preserves the \(O(N)\) symmetry. Accordingly, the functional integral for the non-linear sigma model must be adapted to contain the factor \(\text{det}(G(\vec \pi))\).

Ising model

Everything becomes rather simple again for \(N=1\). The constraint \(\phi(\mathbf{x})^2=2\rho_0\) allows only the field values \(\phi(\mathbf{x})=\pm \sqrt{2\rho_0}\). By a multiplicative rescaling of \(\phi(\mathbf{x})\) one can obtain \(2\rho_0=1\). On a discrete set of space points (a lattice), this leads us to the Ising model.

Gaussian functional integrals and perturbation theory

Gaussian integrals

We now want to develop methods to actually eveluate functional integrals and to calculate correlation functions. We digress for a moment and consider Gaussian integrals of the type \[\int\limits_{\mathbb{R}^N} d^N \varphi \left\{ \exp\left( - \frac{1}{2} \varphi_j K_{jk} \varphi_k + J_k \varphi_k \right) \right\},\] where indices \(j\) and \(k\) are summed in the range \(1, \ldots, N\). The integral is here an infinite volume integral in \(N\) real dimensions weighted by a Gaussian function. We need to assume that the real part of the (symmetric) matrix \(K\) is positive definite, in the sense that the eigenvalues of \(\text{Re}(K_{jk})\) are positive. With some eigenvector \(v_k\) this implies \[v^*_j \text{Re}(K_{jk}) v_k = \text{Re}(v^*_j K_{jk} v_k) = \text{Re}(\lambda) v_k^* v_k >0.\] The source \(J_k\) is not restricted and can be complex. We want to show \[\int\limits_{\mathbb{R}^N} d^N \varphi \left\{ \exp\left( - \frac{1}{2} \varphi_j K_{jk} \varphi_k + J_k \varphi_k \right) \right\} = \frac{(2\pi)^{N/2}}{\sqrt{\text{det} \, K} } \exp\left( \frac{1}{2} J_j (K^{-1})_{jk} J_k \right). \label{eq:01GaussIntegral}\]

Proof

The proof is done in three steps:

  1. Assume first that \(K\) is real and \(J=0\). Then one can find an orthogonal Matrix \(O\) with unit determinant such that \[K = O^T \Lambda O,\] with \(\Lambda = \text{diag}(\lambda_1, \lambda_2, \ldots, \lambda_N)\) a diagomal matrix with real positive entries. One can substitute integration variables \(d^N\varphi \to d^N y\), where \(y_j=O_{jk} \varphi_k\) because the Jacobi determinant is unity here. That implies \[\begin{split} \int d^N \varphi \exp\left(-\frac{1}{2} x_j K_{jk} \varphi_k\right) = \int d^N y \exp\left( -\frac{1}{2} \sum_k \lambda_k y_k^2 \right) = \prod_{k=1}^N \left\{ \lambda_k^{-1/2} \int dx \exp\left( -\frac{1}{2} x^2 \right) \right\}, \end{split}\nonumber\] where we did another variable substitution \(x=\sqrt{\lambda_k}y_k\) in the last step. Now one uses \[\int_{-\infty}^\infty dx \exp\left( -\frac{1}{2} x^2 \right) = \sqrt{2\pi},\] and \[\text{det}(K) = \det(\Lambda) = \prod_{k=1}^N \lambda_k,\] which proves our formula in this special case.

  2. Now consider real \(K\) and real \(J\). Completing the square gives \[\exp\left( - \frac{1}{2} \varphi^T K \varphi + J^T \varphi \right) = \exp\left( \frac{1}{2} J^T K^{-1} J \right) \exp\left( - \frac{1}{2} (\varphi-K^{-1} J)^T K (\varphi-K^{-1} J) \right),\] and the integral over the second term gets reduced to what we have done before with a shift of integration variables.

  3. Finally, the result can be extended to complex \(K\) and complex \(J\) (with the restriction that \(\text{Re}(K)\) has positive eigenvalues) by observing that the left and right hand sides of eq. \(\eqref{eq:01GaussIntegral}\) are holomorphic functions of \(K\) and \(J\).

Gaussian integration can actually be extended to field theories and will be very useful for the following.

Wick theorem

Now that we understand how to do Gaussian integrals we can also consider correlation functions of the type \[G_{ij\cdots k} = \frac{1}{Z} \int\limits_{\mathbb{R}^N} d^N \varphi \left\{ \varphi_i \varphi_j \cdots \varphi_k \exp\left( - \frac{1}{2} \varphi_j K_{jk} \varphi_k \right) \right\},\] First we note that the number of \(\varphi_j\) insertions under the integral must be even, otherwise the integral must yield zero, as a result of the odd transformation behavior of the integrand with respect to reflections, \(\mathbf{\varphi}\to -\mathbf{\varphi}\). To evaluate such integrals we can use the trick \[G_{ij\cdots k} = \left(\frac{1}{Z[J]} \frac{\partial}{\partial J_i} \frac{\partial}{\partial J_j} \cdots \frac{\partial}{\partial J_k} Z[J] \right)_{J=0},\] where \[Z[J] = \int\limits_{\mathbb{R}^N} d^N \varphi \left\{ \exp\left( - \frac{1}{2} \varphi_j K_{jk} \varphi_k + J_k \varphi_k \right) \right\} = \frac{(2\pi)^{N/2}}{\sqrt{\text{det} \, K} } \exp\left( \frac{1}{2} J_j (K^{-1})_{jk} J_k \right)\] is an extended version of the partition function. The prefactor \[\frac{(2\pi)^{N/2}}{\sqrt{\text{det} \, K} }\] cancels out, so all we have to consider is the exponential \[\exp\left( \frac{1}{2} J_j (K^{-1})_{jk} J_k \right).\] Acting now with partial derivative operators brings down terms like \((K^{-1})_{ij}\). Recall that we need to set \(J=0\) at the end.

For example, the two point correlation function gives simply \[G_{ij} = \langle \varphi_i \varphi_j \rangle = (K^{-1})_{ij},\] and similarly, the four-point correlation function gives \[\begin{split} G_{ijkl} = \langle \varphi_i \varphi_j \varphi_k \varphi_l \rangle = & (K^{-1})_{ij} (K^{-1})_{kl} + (K^{-1})_{ik} (K^{-1})_{jl} + (K^{-1})_{il} (K^{-1})_{jk} \\ = & \langle \varphi_i \varphi_j \rangle \langle \varphi_k \varphi_l \rangle + \langle \varphi_i \varphi_k \rangle \langle \varphi_j \varphi_l \rangle + \langle \varphi_i \varphi_l \rangle \langle \varphi_j \varphi_k \rangle. \end{split}\nonumber\] These are examples of a general relation: For a Gaussian (probability) weight one can calculate correlation functions by adding up all possible contractions which each contribute an term of the form of the two-point function or covariance matrix. This is known as Wick’s theorem.

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