Prof. Dr. Stefan Floerchinger

Researcher and professor for quantum field theory.

Classical statistical field theory

Classical field theories are deterministic in the sense that one may fix initial data in the form of the field \(\phi(t,\mathbf{x})\) and its first time derivative or conjugate momentum \(\pi(t,\mathbf{x})= \dot \phi(t, \mathbf{x})\) on some Cauchy surface, and the equation of motion \(\eqref{eq:01eom}\) then fixes it everywhere. However, there are situations where the field configuration is not known precisely, but only stochastic information is available. We will now discuss such situations in some detail. This is interesting by itself but also serves as an excellent technical training for quantum field theory.

Static probabilistic description

Boltzmann probability weights

Recall from statistical mechanics that for a classical system in the canonical ensemble the probability density for a microstate to be in a phase space region \(d^Nq d^Np\) with coordinates \(\mathbf{q}\) and conjugate momenta \(\mathbf{p}\) is given by \[Q(\mathbf{q}, \mathbf{p}) = \frac{1}{Z} \exp\left( - \beta H(\mathbf{q}, \mathbf{p}) \right),\] where \(H(\mathbf{q}, \mathbf{p})\) is the Hamiltonian, \(\beta = 1/(k_\text{B} T)\) is the inverse of temperature times the Boltzmann constant \(k_\text{B}\), and \(Z\) is the partition function. The latter has to be fixed such that the probability distribution is properly normalized, \[Z = \int d^Nq d^Np \exp\left( - \beta H(\mathbf{q}, \mathbf{p}) \right).\] In thermal equilibrium one can determine many different observables by taking averages with respect to the probability distribution \(Q(\mathbf{q}, \mathbf{p})\). For example, the expectation value of energy would be given by \[E = \langle H \rangle = \int d^Nq d^Np \, Q(\mathbf{q}, \mathbf{p}) \, H(\mathbf{q}, \mathbf{p}).\] We are here interested in generalizing this probabilistic description to field theories.

Infinite number of degrees of freedom

When considered from a mechanics point of view, a field theory has one degree of freedom per space point, so formally infinetly many. The infinity arises here in fact for two reasons:

  • Points are dense in space / space is a contiuum.

  • The space \(\mathbb{R}^3\) we consider has infinite volume.

As we will see in due course, these two kinds of infinities lead to all kind of interesting consequences and differences to quantum mechanics for a finite number of degrees of freedom. In order to make progress it is oftentimes needed to regularize the theory. We introduce now a first regularization scheme, although there are many more.

Lattice regularization

A lattice regularization, which is also often used for numerical calculations, consists of two steps:

  • Space is being discretized by considering points on a lattice.

  • The volume is made finite by restricting it to a box, typically with periodic boundary conditions.

To recover the original theory from this regularized theory we need to study two limits:

  • The contiuum limit where the lattice spacing goes to zero.

  • The infinite volume limit where the box size becomes large.

For the technical steps we restric ourselves for simplicity to a single space dimension, the generalization to three spatial dimensions is straightforward.

Consider the chain of points \[x_j \in \{ 0, \varepsilon, 2 \varepsilon, \ldots, (N-1) \varepsilon \},\] with the periodicity condition that \(x_N=N \varepsilon\) is again the point \(x_0=0\). One may visualize this as a ring with circumference \(L=N \varepsilon\). The length \(\varepsilon\) corresponding to the distance between neighboring points is known as the lattice spacing. The contiuum limit corresponds to \(\varepsilon \to 0\), while the infinite volume limit corresponds to \(L\to\infty\).

Let us now consider the real scalar field on this discretized space. The “mechanical” degrees of freedom are essentially the \(N\) field values \[\phi(t, x_j) = \phi(t, j \varepsilon).\] We also need spatial derivatives, which get discretized according to \[\frac{\partial}{\partial x} \phi(t,x) \to \frac{\phi(t,x_{j+1})-\phi(t, x_j)}{\varepsilon}.\]

Discretized Hamiltonian

We leave the details as an excercise and give here just the regularized form of the Hamiltonian for the real scalar field in one spatial dimension, \[H = \sum_{j=1}^{N} \varepsilon \left\{ \frac{1}{2} \pi(t, x_j)^2 + \frac{1}{2} \left( \frac{\phi(t,x_{j+1})-\phi(t,x_j)}{\varepsilon} \right)^2 + V(\phi(t,x_j)) \right\}.\]

Probability distribiution and partition function

For the discretized theory we can immediately write down the thermodynamic equilibrium probability distribution for a field configuration specified by the \(2N\) numbers \(\phi(t,x_j)\), \(\pi(t,x_j)\) with \(j=1,\ldots, N\) at some given time \(t\). In other words, this is a probabilty density for a given field configuration to be in the \(2N\) dimensional infinitesimal phase-space volume element \[\prod_{j=1}^{N} \left\{ d \phi(x_j) d\pi(x_j)\right\},\] and it is given by \[p[\phi, \pi] = \frac{1}{Z} e^{-\beta H}. \label{eq:01probDensity}\] The normalization factor is here the partition function, given as a \(2N\) dimensional integral, \[Z = \prod_{j=1}^{N} \left\{ \int_{-\infty}^\infty d \phi(x_j) \int_{-\infty}^\infty d\pi(x_j)\right\} e^{-\beta H}.\] We introduced here the Boltzmann weight \(e^{-\beta H}\) with (discretized) Hamiltonian \(H\) and inverse temperature \(\beta = 1/T\) (in units where \(k_B=1\)). The partition function is a “sum”, or actually an integral, over the possible field configurations at the given time \(t\) weighted with the Boltzmann factor.

Functional integrals

Functional integral

The functional integral over fields is formally defined from the continuum and infinite volume limit of the phase-space integral \[\int D\phi = \lim_{\varepsilon\to 0, L\to \infty} \prod_{j=1}^{N} \int_{-\infty}^\infty d \phi(x_j),\] and similar for the conjugate momenta. With this we can write the partition function as \[Z = \int D \phi D\pi \, e^{-\beta H[\phi, \pi]}.\] At this point we can at least formally again work with the contiuum version of the field theoretic Hamiltonian \(H\) in \(\eqref{eq:01HamiltonianFieldTheory}\).

A remark is in order at this point: for situations where the Hamiltonian contains terms of higher order than quadratic in the fields (which is the case for the Hamiltonian in \(\eqref{eq:01HamiltonianFieldTheory}\) when \(\lambda > 0\)) the continuum limit needed to define the functional integral is more involved than we have described here. The short distance regularization can only be removed (by letting \(\varepsilon\) go to zero) if the theory is at the same time renormalized. We will discuss renormalization later on. For the time being take the above to be a formal definition of the functional integral.

Expectation values

Of particular interest are observables \(A[\phi]\) that depend on the field \(\phi\) but not the conjugate momenta \(\pi\), for example products of field values at different positions. Such expectation values can then be calculated as \[\langle A[\phi] \rangle = \frac{1}{Z} \int D\phi D\pi A[\phi] \, e^{-\beta H[\phi, \pi]}.\] A first example would be the field expectation value \(\langle \phi(t, \mathbf{x}) \rangle\), another the correlation function of fields at different spatial positions \(\langle \phi(t, \mathbf{x}) \phi(t, \mathbf{y}) \rangle\).

Scaling the partition function

Consider an additative change in the Hamiltonian of the form \[\beta H[\phi, \pi] \to \beta H[\phi, \pi] + C,\] where \(C\) is independent of the fields. This changes the partition function by a factor, \[Z \to e^{-C} Z,\] but does not change expectation values like \(\langle A[\phi] \rangle\) because the factor cancels in the ratio! It can even happen that terms like \(C\) diverge such that formally \(Z\to \infty\) or \(Z\to 0\), but this is not a problem because the absolute value of \(Z\) is irrelvant. The probability density in \(\eqref{eq:01probDensity}\) is not modified by this transformation.

Integrating out the conjugate momenta

Note that the partition function separates into two factors, one involving the conjugate momenta, and one the actual fields. In the discrete version, the functional integral over the conjugate momenta is simply an \(N\)-dimensiona product of Gaussian integrals, \[\prod_{j=1}^N \left\{\int_{-\infty}^\infty d\pi(x_j) e^{-\beta \frac{\varepsilon}{2} \pi(x_j)^2} \right\}.\] These integrals are easily performed and we obtain just a factor \[\left(\frac{2\pi}{\beta \varepsilon}\right)^{N/2}.\] This is in particular independent of the field \(\phi\) and can therefore be dropped according to the argument above. It remains to work with the functioal integral over the actual fields \(\phi\) and a reduced Boltzmann weight where the Hamiltonian involves just the potential energy \(H_\text{pot}[\phi]\).

Euclidean action

The exponent of the Boltzmann weight factor is, mainly for historic reasons, also often called Euclidean action and denoted by \(S[\varphi]\). For example we have for the real scalar field \[S[\varphi] = \beta H_\text{pot} = \int d^3 x \left\{ \frac{1}{2} \partial_j \varphi \partial_j \varphi + V(\varphi) \right\}.\] We have rescaled the fields by a factor, \(\varphi = \sqrt{\beta}\phi\), and adapted the definition of the potential \(V(\varphi)\) accordingy, such that the coefficient of the spatial derivative term becomes \(1/2\). This is a common convention. We are also using Einsteins summation convention where \(j\) is summed from 1 to 3.

As all fields are evaluted at a single instance in time \(t\) we can drop this time argument as long as we are interested in classical thermal equilibrium situations, and work with fields \(\varphi(\mathbf{x})\). In the following we will introduce a somewhat larger class of field theoretic models.

O(N) models

Universality classes and models

In condensed matter physics, microscopic Hamiltonians are often not very well known and if they are, they are not easy to solve. However, in particular in the vicinity of second order phase transitions, there are some universal phenomena that are independent of the precise microscopic physics. This will be discussed in more detail later on, in the context of the renormalization group. Essentially, this arises as a consequence of thermal fluctuations and the fact that at a second order phase transition fluctuations are important on all scales. Roughly speaking, a theory changes in form when fluctuations are taken into account and can approach a largely universal scaling form for many different microscopic theories that happen to be in the same universality class.

In the following we will discuss a class of model systems. These are particularly simple field theories for which one can sometimes answer certain questions analytically, but one can also see them as representatives for their respective universality classes. In the context of quantum field theory, we will see that these field theory models gain a substantially deeper significance.

Scalar O\((N)\) models in \(d\) dimensions

Let us consider models of the form \[S[\phi] =\int d^d x \left\{ \frac{1}{2} \partial_j \phi_n \partial_j \phi_n + \frac{1}{2} m^2 \phi_n \phi_n + \frac{1}{8}\lambda \left(\phi_n \phi_n \right)^2 \right\}. \label{eq:ONModel}\] Here, \(\phi_n = \phi_n(\mathbf{x})\) with \(n=1,\ldots, N\) are the fields. We use Einsteins summation convention which implies that indices that appear twice are summed over. We have formulated the theory in \(d\) spatial dimensions (where in practice \(d=3\), \(2\), \(1\) or even \(0\) for condensed matter systems and \(d=4\) will correspond to a quantum field theory after Wick rotation to Euclidean space). The index \(j\) is accordingly summed in the range \(j=1,\ldots,d\).

Applications

Models of the type \(\eqref{eq:ONModel}\) have many applications. For \(N=1\) they correspond to the continuum limit of the Ising model. For \(N=2\) the model can equivalently be described by complex scalar fields. It has then applications to Bose-Einstein condensates, for example. For \(N=3\) and \(d=3\) one can have situations where the rotation group and the internal symmetry group are coupled. This describes then vector fields, for example magnetization. Finally, for \(N=4\) and \(d=4\), the model essentially describes the Higgs field after a Wick rotation to Euclidean space. Scalar fields are also used in cosmology, for example for the inflaton, or in nuclear physics, for example to describe pions.

Engineering dimensions

In equation \(\eqref{eq:ONModel}\) we have rescaled the fields such that the coefficient of the derivative term is \(1/2\). This is always possible. It is useful to investigate the so-called engineering scaling dimension of the different terms appearing in \(\eqref{eq:ONModel}\). The combination \(\beta H\) or the action \(S\) must be dimensionless. Derivatives have dimension of inverse length \([\partial] = L^{-1}\) and the fields must accordingly have dimension \([\phi] = L^{-\frac{d}{2}+1}\). One also finds \([m]=L^{-1}\) and \([\lambda]=L^{d-4}\). Note in particular that \(\lambda\) is dimensionless in \(d=4\) dimensions.

Symmetries

It is useful to discuss the symmetries of the model \(\eqref{eq:ONModel}\). Symmetries are (almost) always very helpful in theoretical physics. In the context of statistical field theory, they are useful as a guiding principle in particular because they still survive (in a sense to be defined) when the effect of fluctuations is taken into account.

For the model \(\eqref{eq:ONModel}\) we have a space symmetry group consisting of rotations and translations, as well as a continuous, so-called internal symmetry group of global \(O(N)\) transformations. We now discuss them step-by-step.

Rotations

Rotations in space are transformations of the form \[x^j \to x^{\prime j} = R^{jk} x^k.\] The matrices \(R\) fulfill the condition \(R^T R=\mathbb{1}\) and we demand that they connect continuously to the unit matrix \(R=\mathbb{1}\). This fixes \(\det(R)=1\). Matrices of this type in \(d\) spatial dimensions form a group, the special orthogonal group \(SO(d)\). Mathematically, this is a Lie group which implies that all group elements can be composed of many infinitesimal transformations. An infinitesimal transformation can be written as \[R^{jk} = \delta^{jk} + \frac{i}{2}\delta \omega_{mn} \; J_{(mn)}^{jk},\] where \(J_{(mn)}^{jk} = -i(\delta_{mj}\delta_{nk} - \delta_{mk}\delta_{nj})\) are the generators of the Lie algebra and \(\delta\omega_{mn}\) are infinitesimal, anti-symmetric matrices. One may easily count that there are \(d(d-1)/2\) independent components of an anti-symmetric matrix in \(d\) dimensions and as many generators. Finite group elements can be obtained as \[R = \lim_{N\to\infty} \left(\mathbb{1} + \frac{i}{2} \frac{\omega_{mn}}{N} J_{(mn)} \right)^N = \exp\left( \frac{i}{2}\omega_{mn} J_{(mn)} \right).\]

Let us now work out how fields transform under rotations. We will implement them such that a field configuration with a maximum at some position \(\mathbf{x}\) before the transformation will have a maximum at \(R \mathbf{x}\) afterwards. The field must transform as \[\phi_n(\mathbf{x}) \to \phi^\prime_n(\mathbf{x}) = \phi_n(R^{-1} \mathbf{x}).\] Note that derivatives transform as \[\partial_j \phi_n(\mathbf{x}) \to (R^{-1})_{kj} (\partial_k \phi_n)(R^{-1} \mathbf{x}) = R_{jk} (\partial_k \phi_n)(R^{-1}\mathbf{x}).\] The brackets should denote that the derivatives are with respect to the full argument of \(\phi_n\) and we have used the chain rule. The action in \(\eqref{eq:ONModel}\) is invariant under rotations acting on the fields, as one can confirm easily. Colloquially speaking, no direction in space is singled out.

Translations

Another useful symmetry transformations are translations \(\mathbf{x}\to \mathbf{x}+ \mathbf{a}\). The fields get transformed as \[\phi_n(\mathbf{x}) \to \phi^\prime_n(\mathbf{x}) = \phi_n(\mathbf{x}-\mathbf{a}).\] One easily confirms that the action \(\eqref{eq:ONModel}\) is also invariant under translations. Colloquially speaking, this implies that no point in space is singled out.

Global internal \(O(N)\) transformations

There is another useful symmetry of the action \(\eqref{eq:ONModel}\) given by rotations (and mirror reflections) in the “internal” space of fields, \[\phi_n(\mathbf{x}) \to O_{nm} \phi_m(\mathbf{x}).\] The matrices \(O_{nm}\) are here independent of the spatial position \(x\) (therefore this is a global and not a local transformation) and they satisfy \(O^T O=\mathbb{1}\). Because we do not demand them to be smoothly connected to the unit matrix, they can have determinant \(\text{det}(O)=\pm 1\). These matrices are part of the orthogonal group \(O(N)\) in \(N\) dimensions. It is an easy exercise to show that the action \(\eqref{eq:ONModel}\) is indeed invariant under these transformations.

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