# Path integral

Post-publication activity

Curator: Jean Zinn-Justin

A sizable fraction of the theoretical developments in physics of the last sixty years would not be understandable without the use of path or, more generally, field integrals. In this article we will focus on the use of path integrals and field integrals in different branches of theoretical physics. A rigorous study of the mathematical properties of path and field integrals is an open subtopic of functional analysis and will not be dealt with here.

Path integrals are given by sum over all paths satisfying some boundary conditions and can be understood as extensions to an infinite number of integration variables of usual multi-dimensional integrals. Path integrals are powerful tools for the study of quantum mechanics. Indeed, in quantum mechanics, physical quantities can be expressed as averages over all possible paths weighted by the exponential of a term proportional to the ratio of the classical action $$\mathcal S$$ associated to each path, divided by the Planck's constant $$\hbar\ .$$ Thus, path integrals emphasize very explicitly the correspondence between classical and quantum mechanics. In particular, in the semi-classical limit $$\mathcal{S}/\hbar\rightarrow\infty\ ,$$ the leading contributions in the average come from paths close to classical paths, which are stationary points of the action. Thus, path integrals lead to an intuitive understanding and simple calculations of physical quantities in the semi-classical limit.

The formulation of quantum mechanics based on path integrals is well adapted to systems with many degrees of freedom, where a formalism of Schrödinger type is much less useful. Therefore, it allows an easy transition from quantum mechanics to quantum field theory or statistical physics. In particular, generalized path integrals (functional integrals and, more precisely, field integrals) lead to an understanding of the deep relations between quantum field theory and the theory of critical phenomena in continuous phase transitions.

We first describe Brownian motion and Euclidean-time (i.e, imaginary time) path integrals. This means that we consider the path integral representation of the matrix elements of the quantum statistical operator, or density matrix at thermal equilibrium $$\mathrm{e}^{-\beta\hat H},$$ $$\hat H$$ being the quantum Hamiltonian and $$\beta$$ the inverse temperature (measured in a unit where the Boltzmann constant $$k_B$$ is 1). In this way, we are able to describe quantum statistical physics in terms of path integrals, but also, perhaps more surprisingly, to exhibit a relation between classical and quantum statistical mechanics. Moreover, for a whole class of Hamiltonians, the Euclidean-time path integral corresponds to a positive measure. We then define the real-time (in relativistic field theory Minkowskian-time) path integral, which describes the time evolution of quantum systems and corresponds for time-translation invariant systems to the evolution operator $$\mathrm{e}^{-i t\hat H/\hbar}$$ ($$t$$ being the real time). Finally, we briefly list a few generalizations: path integrals in the Hamiltonian formulation, path integrals in the holomorphic representation related to boson systems and, correspondingly, Grassmannian path integrals for fermions.

A number of important applications to physics of the path integral idea involve in fact integrals over fields. In particular, field integrals are indispensable for the study of quantum gauge invariant theories which constitute the basis of the description of fundamental interactions at the microscopic scale, as well as for understanding of the critical properties of phase transitions. They rely on a pragmatic approach, focusing more on developing calculational tools than on establishing rigorous properties. Indeed, even though a number of interesting rigorous results have been proved, one faces extremely difficult mathematical problems in realistic situations (e.g., in four dimensional space-time).

## Random walk, Brownian motion and path integral

As a first example, we consider a random walk on the real line with discrete times $$k=0,1,2,\dots, n.$$ Such a stochastic process is specified by a probability distribution $$P_0(x)$$ for the position $$x$$ at initial time $$k=0$$ and a time-independent density $$\rho(x, x')$$ describing the probability of transition from the point $$x'$$ to the point $$x\ ,$$ meaning that the probability distribution $$P_k(x)$$ at time $$k$$ satisfies the recursion relation or master equation $P_k(x)=\int\mathrm{d}x'\,\rho(x,x')P_{k-1}(x'), \quad \int\mathrm{d} x\,\rho(x,x')=1\,.$ This random walk is a (homogeneous or stationary) Markov chain, that is, a (time-translation invariant) stochastic process with the Markov property (i.e., the property that the probability distribution at time $$k$$ depends only on the probability at time $$k-1\ ,$$ but not on the probability distributions at prior times).

In the following we also assume that the transition probability is translation invariant and even$\rho(x, x')=R(x-x')=R(x'-x)\ .$ Under rather general conditions, the most important being that $$R(x)$$ decreases fast enough for $$|x|$$ large (we call this a local Markov process), one can prove (a consequence of the central limit theorem of probabilities) that the distribution $$P_n(x)$$ converges asymptotically for large times $$n$$ toward a Gaussian distribution that is independent of the transition probability $$R(x)\ .$$ Therefore, if one is interested only in large time properties, one can start directly from a Gaussian transition probability of the form $\tag{1} R(x)={1\over\sqrt{2\pi\xi}}\mathrm{e}^{-x^2/(2\xi)},$

where $$\xi>0$$ characterizes the width of the distribution. It is then easy to calculate $$P_n(x)$$ explicitly by successive Gaussian integrations. However, for our purpose it is more instructive to just apply the recursion relation. If one assumes, for example, that the initial distribution is concentrated at the point $$x=x_0$$ (i.e., $$P_0(x)=\delta(x-x_0)$$ where $$\delta(x)$$ is Dirac's generalized function also known as Dirac function or $$\delta-$$function) one obtains at time $$n$$ the probability distribution $\tag{2} P_n(x,x_0)=\int\mathrm{d} x_{n-1}\mathrm{d} x_{n-2}\ldots \mathrm{d} x_{1}\,R(x-x_{n-1})R(x_{n-1}-x_{n-2})\ldots R(x_1-x_0).$

In the Gaussian example (1), the expression becomes $\tag{3} P_n(x,x_0)=(2\pi\xi)^{(1-n)/2}\int\mathrm{d} x_{n-1}\mathrm{d} x_{n-2}\ldots \mathrm{d} x_{1}\,\mathrm{e}^{-\mathcal{S}(\mathbf{x})/\xi}$, where, defining $$\mathbf{x}\equiv(x_0,x_1,\cdots,x_n)$$ and $$x\equiv x_n\,$$ $\mathcal{S}(\mathbf{x})={1\over2}\sum_{k=1}^{n}\left(x_k-x_{k-1}\right)^2\,$.

We then introduce a time step $$\varepsilon>0\ ,$$ the macroscopic time variables $\tau_k=t'+k\varepsilon\ \mathrm{with}\ 0\le k\le n\,,$ (such that $$\tau_0=t'\ ,$$ $$\tau_n=t'+n\varepsilon\equiv t''$$) and a continuous, piecewise linear path (see Figure 1) $\tag{4} q(\tau)=\sqrt{\varepsilon}\left[ x_{k-1}+{\tau-\tau_{k-1} \over \tau_k-\tau_{k-1}}\left( x_k-x_{k-1}\right)\right]\quad \mathrm{for}\ \tau_{k-1}\le \tau\le \tau_k\quad \mathrm{and}\; k\ge 1,$

with the boundary conditions $\tag{5} q(t')=\sqrt{\varepsilon}x_0\equiv q'\,,\quad q(t'')=\sqrt{\varepsilon} x\equiv q''\,.$

One verifies that $$\mathcal{S}(\mathbf{x})$$ can be rewritten as $\mathcal{S}(\mathbf{x})=\mathcal{S}_\varepsilon(\mathbf{q})\equiv {1\over 2}\int_{t'}^{t''} \,\dot {q}^2(\tau)\mathrm{d} \tau$ where $$\dot{q}(\tau)\equiv \mathrm{d}q/\mathrm{d}\tau\ .$$

One can then study the large discrete time asymptotic behaviour, by taking the large $$n$$ limit at $$t''-t'$$ fixed and, thus, $$\varepsilon=(t''-t')/n\to 0\ .$$ One also speaks of a temporal continuum limit since the time step goes to zero. In this limit, the normalized probability distribution in the new variables $$\Pi_0(t'',t';q'',q')$$ is given by a Euclidean-time path integral (Wiener 1923) that we denote by $\tag{6} \Pi_0(t'',t';q'',q')=\lim_{n\to\infty}{1\over\sqrt{\varepsilon}}P_n(x,x_0)=\int[\mathrm{d} q(\tau)]\mathrm{e}^{-\mathcal{ S}_0(\mathbf{q})/\xi},\quad \mathcal{S}_0(\mathbf{q})={1\over 2}\int_{t'}^{t''} \,\dot{q}^2(\tau)\mathrm{d} \tau$

(the factor $$1/\sqrt{\varepsilon}$$ comes from the change of variables from $$x$$ to $$q$$) where the symbol $$[\mathrm{d} q(\tau)]$$ (also denoted by $$\mathcal{D}q(\tau)$$ in the literature) means sum over all (trajectories) $$q(\tau)$$ satisfying the boundary conditions (5).

### Discussion

A few simple remarks are in order. First, the integrand in the path integral is positive and $$[\mathrm{d} q(\tau)]\mathrm{e}^{-\mathcal{ S}_0(\mathbf{q})/\xi}$$ thus defines a positive measure on paths, the so-called Wiener measure. Second, it is difficult to keep track of the absolute normalization in the continuum path integral limit. Therefore, one mostly uses path integrals to calculate expectation values. If $$\mathcal{F}(\mathbf{q})$$ is a functional of the path $$\mathbf{q}\equiv q(\cdot),$$ its expectation value is defined by $\tag{7} \langle\mathcal{F}(\mathbf{q})\rangle_0 =\mathcal{Z}^{-1}\int[\mathrm{d} q(\tau)]\mathcal{F}(\mathbf{q})\mathrm{e}^{-\mathcal{ S}_0(\mathbf{q})/\xi}$

with $\mathcal {Z}\equiv \int[\mathrm{d} q(\tau)] \mathrm{e}^{-\mathcal{ S}_0(\mathbf{q})/\xi}.$ Note that in the ratio of (7) the normalization cancels. In the case of the random walk, typical expectation values correspond to correlations between positions at different times, which can also be considered as generalized moments of the probability distribution. For example, $\mathcal{F}(\mathbf{q})\equiv q(\tau_1)q(\tau_2)\ldots q(\tau_{2n}).$ Expectation values of this kind are also called correlation functions.

The form of $$\mathcal{ S}_0(\mathbf{q})$$ determines the class of paths that contribute to the path integral, which are called in this case Brownian paths. As the factor $$\sqrt{\varepsilon}$$ in (4) suggests, Brownian paths satisfy a Hölder condition of order 1/2, that is, for $$\tau-\tau'\to0\ ,$$ $| q(\tau)-q(\tau') | = O \left( \left| \tau-\tau' \right|^{1/2} \right);$ in particular Brownian paths are continuous but not differentiable and, thus, $$\dot{q}(\tau)$$ is not defined. In this sense, the notation $$\dot{q}^2(\tau)$$ has to be considered as a symbol and should not taken literally. Nevertheless, it is a useful notation since, for $$\xi\to0\ ,$$ the path integral is dominated by paths close to the classical paths that leave $$\mathcal{ S}_0(\mathbf{q})$$ stationary, and which are differentiable (see the calculation below). Finally, the continuity of the paths allows understanding why it is possible that an integration over an increasingly dense set of points (or sum over all possible piecewise linear paths) eventually converges toward an integration over all continuous paths.

### Explicit calculation

Gaussian path integrals, like finite dimensional Gaussian integrals, are explicitly calculable. Let us illustrate this property with the simple example of the Brownian motion. We now explicitly evaluate the integral (6) by a method that generalizes to other kinds of Gaussian integrals. Varying the quantity $$S_0(\mathbf{q})$$ with respect to the path $$q(\tau),$$ one obtains the classical equation of motion $$\ddot q(\tau)=0\ .$$ Imposing the boundary conditions (5) of the path integral to the classical solution, one obtains $q_c(\tau)=q'+\frac{(\tau-t')}{(t''-t')}(q''-q').$ One then changes variables, $$q(\tau)\mapsto r(\tau)$$ with $$q(\tau)=q_c(\tau)+r(\tau).$$ At each time $$\tau$$ this is a simple translation and the associated Jacobian is $$1\ .$$ The boundary conditions on the new path $$r(\tau)$$ are $$r(t'')=r(t')=0$$ and one finds $S_0(\mathbf{q})=\frac{1}{2}{(q''-q')^2\over(t''-t')}+S_0(\mathbf{r}).$ The path integral (6) becomes $\Pi_0(t'',t';q'',q')=\mathrm{e}^{-\frac{1}{2\xi}{(q''-q')^2\over(t''-t')}} \int[\mathrm{d} r(\tau)]\mathrm{e}^{-\mathcal{ S}_0(\mathbf{r})/\xi}.$ The remaining path integral is $$q'', q'$$-independent and gives then a simple normalization factor. Here, it can be determined by imposing the condition of probability conservation $\tag{8} \int\mathrm{d}q''\,\Pi_0(t'',t'; q'',q')=1\ \Rightarrow\ \Pi_0(t'',t';q'',q')={1\over\sqrt{2\pi\xi(t''-t')}}\mathrm{e}^{-\frac{(q''-q')^2}{2(t''-t')\xi}}.$

As pointed out above, this normalization cancels in expectation values.

## Application of the Wiener measure to statistical physics

The same path integral describing the Brownian motion has an interpretation in the framework of statistical physics.

### Classical statistical physics

The expression (3) in the example (1) may also be physically interpreted as the classical partition function of $$n+1$$ particles on a one-dimensional lattice with spatial sites $$k=0,1,\cdots, n\ .$$ Particles deviate from their equilibrium positions by the value $$x_k$$ and have nearest-neighbour harmonic interactions: $\mathcal {Z}(x_n,x_0;\xi)=(2\pi\xi)^{(1-n)/2}\;\int\mathrm{d} x_{n-1}\mathrm{d} x_{n-2}\ldots \mathrm{d} x_{1} \,\exp\left(-{1\over2\xi}\sum_{k=1}^{n}\left(x_k-x_{k-1}\right)^2\right).$ (The extremities of the chain being fixed at deviations $$x_0$$ and $$x_n\ ,$$ respectively.) Here, the parameter $$\xi$$ has the interpretation of a temperature. The path integral (6) then corresponds to the continuum limit where the lattice spacing $$\varepsilon$$ between two adjacent sites goes to zero at fixed total macroscopic length of the chain $$L=n\varepsilon\ .$$

### Quantum statistical physics

Remarkably enough, the path integral of the Brownian motion yields also the density matrix of a free non-relativistic quantum particle.

The continuum distribution $$\Pi_0(t,t';q,q')$$ given by equation (8) satisfies the diffusion equation ${\partial \Pi_0 \over\partial t}={\xi \over2}{\partial^2 \Pi_0 \over (\partial q)^2} \;,$ with initial condition: $\lim_{t\rightarrow t'}\Pi_0(t,t';q,q')=\delta(q-q')\;.$ This equation can be compared with the equation satisfied by the elements of the quantum density matrix $$\mathrm{e}^{-\beta \hat{H}_0}$$ (in the basis in which the position operator $$\hat q$$ is diagonal $$:\;\hat {q}|q\rangle= q |q\rangle$$), an operator describing the thermal equilibrium of a free non-relativistic quantum particle, where $$\hat{H}_0={\hat{p}}^2/(2 m)$$ is the quantum Hamiltonian in real time (a linear operator acting on the Hilbert space of quantum states), $$\hat p$$ the momentum operator (with the commutation relation $$[\hat q,\hat p]=i\hbar$$), $$m$$ is the particle's mass and $$\beta$$ the inverse temperature (in units of the Boltzmann's constant $$k_\mathrm{B}$$). The matrix elements $$\langle q |\mathrm{e}^{-\beta \hat{H}_0}|q'\rangle$$ satisfy the partial differential equation: ${\partial \over \partial \beta} \langle q | \mathrm{e}^{-\beta \hat{H}_0}|q'\rangle ={\hbar^2\over 2m}{\partial^2 \over (\partial q)^2}\langle q |\mathrm{e}^{-\beta \hat{H}_0}|q'\rangle,$ $$\hbar$$ being Planck's constant. Since $$\Pi_0(t,t';q,q')$$ and $$\langle q | \mathrm{e}^{-\beta \hat{H}_0}|q'\rangle$$ also satisfy the same boundary conditions at initial time $$t-t'\equiv\hbar\beta=0\ ,$$ it follows that $$\langle q |\mathrm{e}^{-\beta \hat{H}_0}|q'\rangle$$ is equal to $$\Pi_0(t=\hbar\beta,0;q,q')$$ when $$\xi=\hbar/m$$ and, therefore, is given by the path integral

$\tag{9} \langle q |\mathrm{e}^{-\beta \hat{H}_0}|q'\rangle =\int[\mathrm{d} q(\tau)]\mathrm{e}^{-\mathcal{ S}_0(\mathbf{q})/\hbar},\quad \mathcal{S}_0(\mathbf{q})={1\over 2}\int_{0}^{\hbar\beta} \,m\dot{q}^2(\tau)\mathrm{d} \tau$

with the boundary conditions $$q(0)=q',$$ $$q(\hbar\beta)=q.$$

## Generalization

A simple generalization of the path integral (9) relevant for quantum statistical physics is the path integral $\tag{10} \Pi(t'',t';q'',q')=\int[\mathrm{d}q(\tau)]\mathrm{e}^{-\mathcal{S}(\mathbf{q})/\hbar} \quad\text{with}\; q(t')=q', q(t'')=q''\;,$

where $\tag{11} \mathcal{S}(\mathbf{q})=\int_{t'}^{t'' }\mathrm{d}\tau\,\mathcal{L}_{\mathrm{E}}(\dot {q}(\tau),q(\tau);\tau)$

and the Euclidean Lagrangian is defined as $\tag{12} \mathcal {L}_{\mathrm{E}}(\dot{q},q;\tau)= \frac{1}{2}m\dot{q}^2+V(q,\tau)\;.$

Note that in the Euclidean Lagrangian the potential $$V(q,\tau)$$ is added to the kinetic energy, while in the normal Lagrangian of classical mechanics the potential is subtracted from the kinetic energy. The parameter $$m$$ can be identified with the mass of a non-relativistic quantum particle. We also assume that $$V(q,t)$$ is a smooth function of $$q$$ and that $\tag{13} \int\mathrm{d}q\, \mathrm{e}^{-\varepsilon V(q,\tau)}<\infty\ \forall \tau \ \mathrm{and}\ \forall \varepsilon>0\,.$

One possible definition of this kind of path integrals refers to the Wiener measure: $\tag{14} \Pi(t'',t';q'',q')=\Pi_0(t'',t';q'',q')\left\langle \exp\left[-{1\over\hbar}\int_{t'}^{t'' }\mathrm{d}\tau\, V(q(\tau),\tau)\right]\right\rangle_0\,,$

where the expectation value is defined in (7) with $$\xi=\hbar/m.$$ With this normalization, $$\Pi(t',t';q'',q')=\delta(q''-q')\ ,$$ which is the kernel associated with the identity operator.

### Path integrals and local Markov processes

We introduce inside the path integral (10) the identity $1=\int\mathrm{d} q\,\delta\bigl(q-q(t)\bigr) \ \mathrm{with}\ t'<t<t'',$ where $$\delta(q)$$ is Dirac's $$\delta$$-function. Also, the expression (11) can be written as the sum $\mathcal{S}(\mathbf{q})=\int_{t'}^t\mathrm{d}\tau\,\mathcal {L}_{\mathrm{E}}(\dot{q},q;\tau) +\int_{t}^{t''}\mathrm{d}\tau\,\mathcal {L}_{\mathrm{E}}(\dot{q},q;\tau).$ The path integral thus factorizes into the product of two path integrals with boundary conditions at $$t''$$ and $$t\ ,$$ and $$t$$ and $$t'\ ,$$ respectively, integrated over the intermediate point $$q\ .$$ One concludes that $$\Pi(t'',t';q'',q')$$ defined by the path integral (10) satisfies a Markov property in (Euclidean) time: $\tag{15} \Pi(t'',t';q'',q')=\int\mathrm{d}q\; \Pi(t'',t;q'',q)\; \Pi(t,t';q,q')\quad \mathrm{for}\ t''\ge t\ge t' \,.$

This multiplication rule also shows that $$\Pi(t'',t';q'',q')$$ can be identified with the kernel or matrix element of an operator $$\boldsymbol{\Pi}(t'',t')$$ in an Hilbert space $$\mathcal{H}\ .$$ In Dirac's bra-ket notation, $\tag{16} \langle q''|\boldsymbol{\Pi}(t'',t')|q'\rangle\equiv \Pi(t'',t';q'',q').$

In operator notation, the relation (15) becomes $\boldsymbol{\Pi}(t'',t')=\boldsymbol{\Pi}(t'',t)\boldsymbol{\Pi}(t,t').$ The Markov property (15) allows writing the path integral as a product of $$n$$ path integrals integrated over intermediate points, corresponding to a time interval $$\varepsilon=(t''-t')/n$$ that can be chosen arbitrarily small by increasing $$n\ .$$ The path integral can thus be evaluated from its asymptotic behaviour for small time intervals. Then, with the boundary conditions $$q(t)=q,$$ $$q(t+\varepsilon)=q_\varepsilon\ ,$$ $\tag{17} \mathcal{S}(\mathbf{q})=\int_{t}^{t+\varepsilon}\mathrm{d}\tau\left[\frac{1}{2}m\dot{q}^2(\tau)+V(q(\tau),\tau)\right]\sim m{(q_\varepsilon-q)^2\over2\varepsilon}+\varepsilon V(q,t).$

For a given trajectory, the leading term in the expression when $$\varepsilon\rightarrow0$$ is still the Brownian term (kinetic term in classical mechanics), which implies that the paths contributing to the path integral still satisfy the Hölder property $$|q_\varepsilon-q|=O(\varepsilon^{1/2})\ .$$ In particular, this property implies that the argument $$q$$ of $$V$$ in (17) can be replaced by any value linearly interpolating between $$q$$ and $$q_\varepsilon\ ,$$ the difference being of order $$\varepsilon^{1/2}$$ hence negligible in this approximation. The dominance of the Brownian term implies also a (spatial) locality property (and thus the denomination of local Markov process): for $$\varepsilon$$ small, $$\langle q_\varepsilon|\boldsymbol{\Pi}(t+\varepsilon,t)|q\rangle$$ decreases exponentially when $$|q-q_\varepsilon|\to \infty\ .$$ Since at leading order in $$\varepsilon$$ the normalization is provided by the Brownian motion, one concludes that $\tag{18} \langle q_\varepsilon|\boldsymbol{\Pi}(t+\varepsilon,t)|q\rangle\sim \Pi_\varepsilon(t;q_\varepsilon,q)=\sqrt{m\over2\pi\hbar\varepsilon} \exp\left[-\left(m{(q_\varepsilon-q)^2\over2\varepsilon}+\varepsilon V(q,t)\right)/\hbar\right] .$

This leads to an alternative definition of the path integral (10) as the limit when $$n\to\infty\$$ at $$n\varepsilon=t''-t'$$ fixed of an $$n$$-dimensional integral: $\tag{19} \langle q'' | \boldsymbol{\Pi} (t'' ,t' ) | q' \rangle =\lim_{n\to \infty} \int \prod ^{n-1}_{k=1}\mathrm{d} q_{k} \prod^{n}_{k=1} \Pi_\varepsilon(\tau_{k-1};q_k,q_{k-1})$

with the conventions $$\tau_{k}=t' +k\varepsilon\ ,$$ q$$_{0}=q'\ ,$$ $$q_{n}=q ''\ .$$ At finite $$n$$ the right hand side defines a Markov process in discrete times.

### Path integrals and classical statistical physics

If the interpretation of the path integral in terms of random walks (see (14)) and Markov processes is somewhat indirect, the interpretation in the framework of classical statistical physics is simple. In the form of the right hand side of (19), the path integral appears as the continuum limit of the classical partition function of a one-dimensional lattice model. The configuration energy of the lattice model is (here we set $$m=\hbar=1$$) $S(\mathbf{q})=\sum_{k=1}^{n}\left[{1\over2\varepsilon}\left(q_k-q_{k-1}\right)^2+\varepsilon V(q_k)\right]\,,$ where we have assumed $$V$$ time independent to enforce translation invariance. The continuum limit $$\varepsilon\propto 1/ n \to 0$$ corresponds also to a kind of low temperature limit where the correlation length in the statistical model, which is proportional to $$1/\varepsilon\ ,$$ diverges.

### Path integrals and quantum statistical physics

From the evaluation of the path integral at short time intervals, one can also infer that $$\langle q'' | \boldsymbol{\Pi} (t'' ,t' ) | q'\rangle$$ satisfies the partial differential equation $\tag{20} \hbar {\partial \over \partial t}\langle q | \boldsymbol{\Pi} (t ,t' ) | q'\rangle=\left[{\hbar^2\over 2m}{\partial^2\over (\partial q)^2}-V(q,t)\right] \langle q | \boldsymbol{\Pi} (t ,t' ) | q'\rangle\;,$

with the initial condition$$\langle q | \boldsymbol{\Pi}(t' ,t') | q'\rangle =\delta (q-q')$$ (i.e., $$\boldsymbol{\Pi} (t',t' ) = \mathbf{1}$$).

When the potential is time-independent, that is, when $$V(q,t)=V(q)\ ,$$ equation (20) is related by the formal substitution $$t\mapsto it$$ to the Schrödinger equation in real time: $\tag{21} i\hbar{\partial \over \partial t }\langle q | \mathbf{U} (t ,t' ) | q'\rangle=\left[-{\hbar^2\over 2m}{\partial^2\over (\partial q)^2}+V(q)\right] \langle q | \mathbf{U} (t ,t' ) | q'\rangle\,,$

for the matrix elements of the quantum evolution operator $$\mathbf{U} (t'' ,t' ).$$ The solution of (21) with initial condition $$\mathbf{U}(t' ,t' )=\mathbf{1}$$ is $\tag{22} \mathbf{U}(t ,t')= \mathrm{e}^{-i (t -t' )\hat{H}/\hbar} \;,$

where $\tag{23} \hat{H}=\frac{{\hat p}^2}{2 m}+V(\hat{q})\;,$

is a time-independent quantum Hamiltonian and $$\hat q$$ and $$\hat p$$ are, respectively, the position and momentum operators. Note also that with the condition (13), the Hamiltonian (23) has a discrete spectrum.

Equation (20), also called imaginary time Schrödinger equation, has then a solution $$\boldsymbol{\Pi}(t,t') =\mathrm{e}^{- (t -t' )\hat{H}/\hbar}$$ with initial condition $$\boldsymbol{\Pi}(t,t)=\mathbf{1}$$ and, therefore, $\tag{24} \boldsymbol{\Pi}(t=\beta\hbar,0)= \mathrm{e}^{-\beta \hat{H}}\,,$

where $$\mathrm{e}^{-\beta \hat{H}}$$ is the density matrix operator describing the thermodynamic equilibrium of a quantum system with quantum Hamiltonian $$\hat{H}$$ at temperature $${1}/{\beta}\ .$$ Equations (10), (11), (16) and (24) generalize (9).

The partition function $$\mathcal{Z}(\beta)=\mathrm{tr} \,\mathrm{e}^{-\beta\hat H}=\mathrm{tr}\boldsymbol{\Pi}(\hbar\beta/2,-\hbar\beta/2)$$ of the quantum system can be represented as $\tag{25} \mathcal{Z}(\beta)=\int \mathrm{d}q \,\Pi(\hbar\beta/2,-\hbar\beta/2;q,q) =\int[\mathrm{d}q(\tau)]\,\mathrm{e}^{-\mathcal{S}(\mathbf{q})/\hbar} \qquad\text{sum over all }\; q(\hbar\beta/2)=q(-\hbar\beta/2) \;,$

that is as a path integral integrating over all closed paths, that is, all paths satisfying the periodic boundary condition $$q(\hbar\beta/2)=q(-\hbar\beta/2).$$

## Gaussian path integrals: The quantum harmonic oscillator

Gaussian path integrals, like finite dimensional Gaussian integrals, can be calculated explicitly. A more general example is provided by the path integral representation associated to the quantum harmonic oscillator defined by the quantum Hamiltonian $\tag{26} \hat H=\frac{{\hat p}^2}{2 m}+\textstyle{\frac{1}{2} }m\;\omega^2\; {\hat{q} }^2\;,$

where $$m$$is the mass of the particle and $$2\pi/\omega$$ is the period of the classical oscillations. Then, the expression (12) becomes $\mathcal {L}_{\mathrm{E}}(\dot{q},q;\tau)= \textstyle{\frac{1}{2}}m\left(\dot{q}^2+ \omega^2 q^2\right).$ Like in the example of the Brownian motion, to calculate the corresponding path integral one first solves the classical equation of motion (Euler-Lagrange equation corresponding to $$\mathcal {L}_{\mathrm{E}}$$) $-\ddot q(\tau)+\omega^2q(\tau)=0$ with the path integral boundary conditions $$q(t/2)=q''\ ,$$ $$q(-t/2)=q' .$$ (Since the potential is time-independent, we can choose $$t'=-t/2$$ and $$t''=t/2$$ in (10).) The classical solution is $q_c(\tau) ={1\over\sinh (\omega t)}\left[q'\sinh\bigl(\omega(t/2-\tau)\bigr)+ q'' \sinh\bigl(\omega(\tau+t/2)\bigr)\right]$ and the corresponding classical action is $\mathcal{S}(\mathbf{q_c})={m\omega \over 2 \sinh \omega t}\left[ \left(q^{\prime 2}+q^{\prime\prime 2} \right) \cosh \omega t -2q' q'' \right] .$ One then changes variables $$q(\tau)\mapsto r(\tau)=q(\tau)-q_c(\tau)\ ,$$ such that $$r(-t/2)=r(t/2)=0\ .$$ Since $\mathcal{S}(\mathbf{q})=\mathcal{S}(\mathbf{q_c})+\mathcal{S}(\mathbf{r}),$ the path integral becomes $\langle q''|\boldsymbol{\Pi}(t/2,-t/2)|q'\rangle=\mathrm{e}^{- \mathcal{S}(\mathbf{q_c})/\hbar}\int[\mathrm{d}r(\tau)]\mathrm{e}^{-\mathcal{S}(\mathbf{r})/\hbar},$ The integral over $$r(\tau)$$ yields a normalization constant. Its dependence on $$\omega$$ can be determined by differentiating the path integral with respect to $$\omega\ .$$ The final normalization can be determined by comparing with the Brownian motion $$\omega=0.$$ One finds $\int[\mathrm{d}r(\tau)]\mathrm{e}^{-\mathcal{S}(\mathbf{r})/\hbar}=\left({ m\omega \over 2\pi \hbar\, \sinh \omega t }\right)^{1/2}.$ The quantum partition function $$\mathcal{Z}(\beta)=\mathrm{tr} \,\mathrm{e}^{-\beta\hat H}$$ (see (25)) follows: $\tag{27} \mathcal{Z}(\beta)=\int\mathrm{d} q\,\langle q|\boldsymbol{\Pi}(\hbar\beta/2,-\hbar\beta/2)|q\rangle={1\over2\sinh(\beta\hbar\omega/2)} ={\mathrm{e}^{-\beta\hbar\omega/2} \over 1-\mathrm{e}^{-\beta\hbar\omega}}=\sum_{n=0}^{\infty}\mathrm{e}^{-\beta\hbar\omega(n+\frac{1}{2})}\;.$

From (27) and the definition $$\mathcal{Z}(\beta)=\mathrm{tr} \,\mathrm{e}^{-\beta\hat H}\ ,$$ one recovers the exact spectrum of the quantum Hamiltonian $$\hat H$$ whose eigenvalues are $$E_n=\hbar\omega(n+\textstyle{\frac{1}{2}})\ .$$

## Perturbative expansion for the path integral

### Correlation functions

Generalizing to the interacting case the definition given for the free particle in the section "Discussion", correlation functions are the moments of the measure associated to the integrand in the path integral. They are defined by $\tag{28} \langle q(\tau_1)q(\tau_2)\ldots q(\tau_{p})\rangle={1\over\mathcal{Z}}\int[\mathrm{d}q(\tau)]q(\tau_1)q(\tau_2)\ldots q(\tau_{p})\mathrm{e}^{-\mathcal{S}(\mathbf{q})/\hbar}$

with $\mathcal{Z}=\int[\mathrm{d}q(\tau)]\mathrm{e}^{-\mathcal{S}(\mathbf{q})/\hbar}.$ In the example of periodic boundary conditions $$q(-t/2)=q(t/2),$$ ($$q(t/2)$$ integrated), which correspond to a summation over all closed trajectories, $$\mathcal{Z}(\beta)$$ is the quantum partition function describing the thermodynamical equilibrium at temperature $$1/\beta=\hbar/t\ .$$

### Gaussian expectation values and Wick's theorem

We now consider path integrals corresponding to centred Gaussian measures, for which the action is a quadratic form in terms of the integration path $$q(\tau),$$ (simple examples being provided by the Brownian motion and the quantum harmonic oscillator). Then, the symmetry $$q\mapsto -q$$ implies that correlation functions (28) odd in $$q$$ vanish. Moreover, for all centred Gaussian measures, correlation functions can be expressed in terms of the two-point function as stated by Wick's theorem (proved in a different context by Wick (1950)): $\langle q(\tau_1)q(\tau_2)\ldots q(\tau_{2\ell}) \rangle=\sum _{ \scriptstyle{P \left\{ 1,2,\ldots 2\ell \right\}}}\langle q(\tau_{P_1})q(\tau_{P_2})\rangle\ldots \langle q(\tau_{P_{2\ell-1}}) q(\tau_{P_{2\ell}})\rangle,$ where $$P \left\{ 1,2,\ldots, 2\ell \right\}$$ are all possible (unordered) pairings of $$\left\{ 1,2,\ldots, 2\ell \right\}\ .$$ For example, $\langle q(\tau_1)q(\tau_2)q(\tau_3)q(\tau_4)\rangle=\langle q(\tau_1)q(\tau_2)\rangle \langle q(\tau_3)q(\tau_4)\rangle +\langle q(\tau_1)q(\tau_3)\rangle \langle q(\tau_2)q(\tau_4)\rangle +\langle q(\tau_1)q(\tau_4)\rangle \langle q(\tau_3)q(\tau_2)\rangle.$ One proof of the theorem relies on introducing the generating functional of correlation functions $\mathcal{Z}(j(\tau))=\langle \exp \int\!\mathrm{d}\tau \; j(\tau) q(\tau)\rangle,$ where $$j(\tau)$$ is an arbitrary function. Correlations functions are obtained by taking multiple functional derivatives of $$\mathcal{Z}(j(\tau))$$ with respect to $$j(\tau)\ .$$ The generating functional $$\mathcal{Z}(j(\tau))$$ can be calculated explicitly by Gaussian integration. Its expansion in powers of $$j(\tau)$$ leads to Wick's theorem.

In the example of the quantum harmonic oscillator and periodic boundary conditions, in the limit $$t\to\infty$$ the two-point function reduces to $$\langle q(\tau_1)q(\tau_2)\rangle \sim{\hbar\over2 m\omega} \mathrm{e}^{-\omega|\tau_1-\tau_2|}.$$

### Perturbative expansion

We now assume that the potential $$V(q,t)$$ in the path integral is a polynomial in $$q\ ,$$ though perturbation theory can be generalized to analytic potentials. In the section Gaussian expectation values and Wick's theorem it has been shown that Gaussian expectation values can be calculated explicitly: therefore, to evaluate a path integral, a possible method is to keep the quadratic part ($$O(q^2)$$) of the potential of $$V(q,t)$$ in the exponential as part of the Gaussian measure and to expand the remainder in a power series. For illustration purpose, we consider the quartic anharmonic oscillator, $V(q)=\frac{1}{2} q^2+ \lambda q^4, \lambda >0\,.$ We set $\mathcal{S}(\mathbf{q})=\mathcal{S}_0(\mathbf{q})+\lambda\int\mathrm{d}\tau \,q^4(\tau)$ with $\mathcal{S}_0(\mathbf{q})=\frac{1}{2}\int\mathrm{d}\tau\,\bigl(\dot{q}^2(\tau)+q^2(\tau)\bigr).$ The expansion of the corresponding path integral can then be written as (here we set $$\hbar=1$$) $\mathcal{Z}(\lambda)= \int[\mathrm{d}q(\tau)]\mathrm{e}^{-\mathcal{S}_0(\mathbf{q})}\sum_{k=0}^\infty {(-\lambda)^k\over k!}\left[\int\mathrm{d}\tau \,q^4(\tau)\right]^k\sim\mathcal{Z}(0)\sum_{k=0}^\infty {(-\lambda)^k\over k!}\left\langle \left[\int\mathrm{d}\tau \,q^4(\tau)\right]^k\right\rangle_0 ,$ where $$\langle\bullet\rangle_0$$ means expectation value with respect to the Gaussian measure associated with $$\mathcal{S}_0$$ and the symbol $$\sim$$ means that the perturbative series is not convergent. Each term in the series can then be evaluated using Wick's theorem and the explicit form of the Gaussian two-point function. For example, at first order in $$\lambda\ ,$$ $\langle q^4(\tau)\rangle_0=3 (\langle q^2(\tau)\rangle_0)^2.$ The next order involves $\langle q^4(\tau)q^4(\tau')\rangle_0=9 (\langle q^2(\tau)\rangle_0)^2(\langle q^2(\tau')\rangle_0)^2+72 \langle q^2(\tau)\rangle_0 \langle q^2(\tau')\rangle_0(\langle q(\tau)q(\tau')\rangle_0)^2+24 (\langle q(\tau)q(\tau')\rangle_0)^4.$ It is then convenient to represent individual contributions graphically in terms of Feynman diagrams. Let us point out that such an expansion is divergent for all values of the parameter $$\lambda\ .$$ It is an asymptotic series, useful as such only for $$\lambda$$ small enough. For larger values of the expansion parameter, series summation methods are required.

## Quantum time evolution

Following Feynman (Feynman 1948), quantum time-evolution (here we refer to real physical time) can be described in terms of (oscillatory) path integrals. In this formalism, considering a system classically described by the Cartesian coordinates $$\mathbf{q}\equiv\{q^1,q^2\ldots\}\ ,$$ the matrix elements of the quantum evolution operator $$\mathbf{U}(t'',t')$$ between times $$t'$$ and $$t''$$ are given by a sum over all possible trajectories (paths) $$\mathbf{q}(\tau)\equiv\{q^1(\tau),q^2(\tau)\ldots\}\ ,$$ which in the simplest cases can be written as $\tag{29} \langle \mathbf{q}'' \left|\mathbf{U}(t'',t') \right| \mathbf{q}' \rangle = \int \left [ \mathrm{d} \mathbf{q} (\tau) \right] \exp\left({i \over \hbar}\mathcal{A} (\mathbf{q} )\right)$

with the boundary conditions $\tag{30} \mathbf{q}(t')=\mathbf{q}' , \ \mathbf{q}(t'')=\mathbf{q}'',$

where the classical action $$\mathcal{A} (\mathbf{q} )$$ is the time-integral of the classical Lagrangian: $\tag{31} \mathcal{A}( \mathbf{q})=\int_{t'}^{t''}\mathrm{d} \tau\, \mathcal{L}\left( \mathbf{q}(\tau),\dot{\mathbf{q}}(\tau);\tau\right).$

The expression (29) is valid when the kinetic term, that is, the term with two time-derivatives in the Lagrangian has the form $$\textstyle{\frac{1}{2}\sum_i} m_i(\dot{q}^i)^2 \ ,$$ otherwise the measure has to be modified and new problems arise. An example of the latter situation is provided when the coordinates $$q^i$$ parametrize a Riemannian manifold and the kinetic term involves $$\textstyle{\sum_{i,j}}\dot{q}^i g_{ij}(\mathbf{q})\dot{q}^j\ ,$$ where $$g_{ij}$$ is the metric tensor.

The formulation of quantum mechanics in terms of path integrals actually explains why equations of motion in classical mechanics can be derived from a variational principle. In the classical limit, that is, when the typical classical action is large with respect to $$\hbar\ ,$$ the path integral can be evaluated by using the stationary phase method. The sum over paths is thus dominated by paths that leave the action stationary: the classical paths that satisfy $\tag{32} \mathcal{A}\left(\mathbf{q}+\delta\mathbf{q}\right)-\mathcal{A}\left(\mathbf{q}\right)=O(\|\delta\mathbf{q}\|^2)\ \Rightarrow\ \frac{\delta \mathcal{A}}{\delta q^i}=0\ \Rightarrow\ {\partial\mathcal{ L}\over\partial q^i}-{\mathrm{d} \over\mathrm{d} t}{\partial\mathcal{L} \over\partial \dot{q}^i}=0$

with the boundary conditions (30). The leading order contribution is then obtained by expanding the path around the classical path, keeping only the quadratic term in the deviation and performing the corresponding Gaussian integration.

This property generalizes to relativistic quantum field theory.

From the mathematical point of view, it is much more difficult to define rigorously the real-time path integral than the imaginary-time statistical path integral. A possible strategy involves, when applicable, to calculate physical observables for imaginary time and then to proceed by analytic continuation.

## Barrier penetration in the semi-classical limit

The purpose of this section is to illustrate with a simple example the evaluation of statistical (or imaginary time) path integrals in the semi-classical approximation. It is more technical and can be omitted in a first reading.

The path integral associated with $$\mathrm{tr}\,\mathrm{e}^{-t\hat H/\hbar}\ ,$$ $\tag{33} \mathcal{Z}=\int[\mathrm{d}q(\tau)]\mathrm{e}^{-\mathcal{S}(\mathbf{q})/\hbar} \quad\text{with}\; q(t/2)=q(-t/2),$

is especially well suited to the evaluation, in the semi-classical limit $$\hbar\to 0\ ,$$ of specific quantum phenomena called barrier penetration or tunnelling. Indeed, it can be shown that the classically forbidden barrier penetration appears, in the semi-classical limit, as formally related to classical evolution in imaginary time.

To explain the general idea, we consider an example of the form $\tag{34} \mathcal{S}(\mathbf{q})=\int_{-t/2}^{t/2 }\mathrm{d}\tau\left[\frac{1}{2}m\dot{q}^2(\tau)+V\bigl(q(\tau)\bigr)\right],$

with the potential $\tag{35} V(q)=\frac{1}{2}q^2-\frac{1}{2}\lambda q^3\,,$

which has one local minimum at $$q=0$$ ($$V=0$$), a local maximum at $$q=2/(3\lambda)$$ ($$V=4/(27\lambda^2)$$) and goes to $$\mp\infty$$ when $$q\to \pm\infty\ .$$

The problem is to evaluate the probability per unit time for a particle localized initially in the well of the potential at $$q=0$$ to escape the well. Since the potential (35) is not bounded from below, it is first necessary to define the quantum Hamiltonian. In this example, one can proceed by analytic continuation starting from $$\lambda$$ pure imaginary. As conjectured initially by Bessis and Zinn-Justin (unpublished), the corresponding Hamiltonian, though complex, has a discrete real spectrum as a consequence of the symmetry $$q\mapsto-q\ ,$$ $$\hat H\mapsto \hat H^*\ ,$$ a symmetry also called PT symmetry (P being the parity transformation and T the time-reversal transformation). Returning by analytic continuation to $$\lambda$$ real, one finds in this case a complex energy spectrum (quantum resonances), the imaginary part of the energy eigenvalues being directly related to tunnelling.

In the path integral framework, it can be shown that barrier penetration effects can be derived from an evaluation of the integral (33) for $$\hbar\to0$$ and, therefore, by the steepest-descent method, suitably generalized to path integrals. One looks for non-trivial saddle points, here non-constant solutions of the classical equations of motion derived from the Euclidean action (34), which correspond formally to evolution in imaginary time. Moreover, if one is interested only in states with energies of order $$\hbar\ ,$$ then one has to take the limit $$|t|\to\infty\ .$$ Therefore, one looks for solutions that have a finite action on the real line. These solutions are called instantons. Here, the equation of motion obtained by varying $$\mathcal{S}$$ is $-\ddot q(\tau)+q(\tau)-\frac{3}{2}\lambda q^2(\tau)=0\,.$ Due to time translation invariance, one finds a one-parameter family of instanton solutions: $q_c(\tau)={1\over\lambda \cosh^2((\tau-\tau_0)/2)}\ \Rightarrow\ \mathcal{S}(\mathbf{q}_c)=\int_{-\infty}^{+\infty}\mathrm{d}\tau \,\mathcal{L}_{\mathrm{E}}(\mathbf{q}_c)={8\over 15\lambda^2}.$ Completing the calculation of the saddle point contribution is a non-trivial exercise because it requires factorizing the path integral measure into an integration over $$\tau_0$$ (a collective coordinate related to the breaking of time-translation symmetry by the solution) before using a saddle point approximation for the other modes of the path. One infers that, up to power law corrections, the probability per unit time of leaving the well is of order $$\exp{-\mathcal{S}(\mathbf{q}_c)/\hbar}=\exp{-8/(15\lambda^2\hbar)}.$$

## Path integrals: Generalizations

We have presented only the simplest form of path integrals, which for the point of view of quantum mechanics involve only a classical Lagrangian with the general form (11). For more general Lagrangians or Hamiltonians, one encounters new problems in the definition of path integrals.

### The quantum particle in a static magnetic field

When the Lagrangian involves a term linear in the velocity, as in the example of a quantum particle in a magnetic field, $\tag{36} \mathcal{L}( \mathbf{q},\dot{\mathbf{q}})=\textstyle{1\over2 } \, m \,\dot{\mathbf{q}}^2-e \, \mathbf{A}( \mathbf{q})\cdot \dot {\mathbf{q}}\, ,$

where $$\mathbf{A}( \mathbf{q})$$ is a given vector potential, a new problem related to quantization arises. The classical Lagrangian together with the correspondence principle (replacing position and velocity by the corresponding quantum operators) does not determine the quantum theory because operators $$\mathbf{A}( \hat{\mathbf{q}})$$ and $$\dot{\hat{\mathbf{q}}}$$ no longer commute. Correspondingly, the naive continuum form of the path integral is not defined because the continuum limit depends explicitly on the time-discretized form of the path integral and leads to a one-parameter family of different theories. This reflects, for example, in the appearance of undefined terms $$\operatorname{sgn}(0)$$ in calculations. The underlying quantum Hamiltonian is then uniquely determined by demanding either its hermiticity or equivalently its gauge invariance. To determine the path integral, one can either return to a time-discretized form consistent with the quantum Hamiltonian (which implies the midpoint rule in the argument of the vector potential), or add a term with higher order time derivatives in the action, for example, $\mathcal{S}\mapsto \mathcal{S}+\eta\int_{t'}^{t''}\mathrm{d}\tau(\ddot q(\tau))^2, \ \eta>0\,.$ This has the effect of restricting the integration to paths that satisfy a Hölder condition of order 3/2 and are thus differentiable, in such a way that expectations values with $$\dot{\mathbf{q}}$$ are defined. This regularization does not violate gauge invariance but violates hermiticity of the Hamiltonian. In the case of the magnetic field, in the $$\eta\to0$$ limit, it fixes the ambiguities ($$\operatorname{sgn}(0)=0$$) in a way that is consistent with a gauge invariant Hermitian quantum Hamiltonian.

### Hamiltonian formulation and phase space integration

For a general classical Hamiltonian, the quantum evolution operator can formally be expressed in terms of a path integral involving an integration over phase space variables, position $$\mathbf{q}$$ and conjugate momentum $$\mathbf{p}\ :$$ $\tag{37} \langle \mathbf{q}'' \left| \mathbf{U}(t'',t') \right| \mathbf{q}' \rangle = \int \left [ \mathrm{d} \mathbf{p} (\tau) \mathrm{d} \mathbf{q} (\tau)\right] \exp\left({i \over \hbar}\mathcal{A} (\mathbf{p},\mathbf{q} )\right)$

with the boundary conditions $$\tag{38} \mathbf{q}(t')=\mathbf{q}' , \ \mathbf{q}(t'')=\mathbf{q}'',$$

where the classical action $$\mathcal{A} (\mathbf{p},\mathbf{q} )$$ is now expressed in terms of the classical Hamiltonian $$H(\mathbf{p},\mathbf{q};t):$$ $\tag{39} \mathcal{A}(\mathbf{p},\mathbf{q})=\int_{t'}^{t''}\mathrm{d} \tau\,\left[\mathbf{p}(\tau)\cdot\dot{\mathbf{q}}(\tau)- H\!\left(\mathbf{p}(\tau), \mathbf{q}(\tau); \tau\right)\right].$

When the Hamiltonian is quadratic in the conjugate momentum $$\mathbf{p},$$ the integral over $$\mathbf{p}(\tau)$$ is Gaussian and can be performed explicitly: first one shifts $$\mathbf{p}(\tau)$$ in the exponential by the solution of the classical equation $$\dot{\mathbf{q}}(\tau)=\frac{\partial H\!\left(\mathbf{p}(\tau), \mathbf{q}(\tau); \tau\right) }{\partial\mathbf{p}(\tau) } \ .$$ One thus recovers in the exponential the classical Lagrangian. One then integrates over $$\mathbf{p}(\tau)$$ and this may modify the $$\mathbf{q}(\tau)$$-integration measure if the coefficient of the quadratic term in $$\mathbf{p}(\tau)$$ is not a constant. In the general case, the interpretation of this path integral reflects the problems of quantizing classical Hamiltonians and the order of operators in products. The Hamiltonian path integral has mainly a heuristic value (except in the semi-classical limit).

### Holomorphic formalism and bosons

Up to now, we have described the path integral formalism relevant for distinct quantum particles. But quantum particles are either bosons, obeying the Bose-Einstein statistics or fermions, governed by Fermi-Dirac statistics. To describe the quantum evolution of several identical (and thus indiscernible) quantum particles, the path integral formulation has to be generalized. In the case of bosons, it is based on the coherent states holomorphic formalism and the Hilbert space of analytic entire functions. For bosons occupying only a finite number of quantum states, the relevant path integral can formally be deduced from the phase space integral by a complex change of variables, up to boundary terms and boundary conditions. In the example of one quantum state, the change of variables is simply $z=(p+iq)/i\sqrt{2}\,,\quad \bar z=i(p-iq)/\sqrt{2}\,.$ The holomorphic path integral then takes the form $\tag{40} \langle \mathbf{z}'' \left|\mathbf{U} (t'',t') \right| \bar{\mathbf{z}}' \rangle = \int \left [ \mathrm{d}\bar {\mathbf{z}} (\tau) \mathrm{d} \mathbf{z} (\tau)\right]\mathrm{e}^{\bar{\mathbf{z}}(t')\cdot\mathbf{z}(t')} \exp\left({i \over \hbar}\mathcal{A} (\mathbf{z},\bar{\mathbf{z}} )\right)$

with the boundary conditions $$\tag{41} \bar{\mathbf{z}}(t')=\bar{\mathbf{z}}' , \ \mathbf{z}(t'')=\mathbf{z}'',$$

where the classical action $$\mathcal{A} (\mathbf{z},\bar{\mathbf{z}} )$$ reads $\tag{42} \mathcal{A}(\mathbf{z},\bar{\mathbf{z}})=\int_{t'}^{t''}\mathrm{d} \tau\,\left[-i\bar{\mathbf{z}}(\tau)\cdot\dot{\mathbf{z}}(\tau)- H\left(i(\mathbf{z}(\tau)-\bar {\mathbf{z}}(\tau))/\sqrt{2} ,(\mathbf{z}(\tau)+\bar{\mathbf{z}}(\tau))/\sqrt{2} ; \tau\right)\right].$

More generally, to $$N$$ quantum states are associated $$N$$ pairs of complex variables $$(z_i,\bar{z_i})\ .$$

Even in the Gaussian example, this path integral suffers from the same ambiguities as in the example of a particle in a magnetic field, and this leads also to the appearance of $$\operatorname{sgn}(0)$$ in calculations.

### Grassmann path integrals and fermions

The understanding of this section necessitates some prior knowledge of Grassmann or exterior algebras, including the definition and properties of Grassmann differentiation and integration.

The description of the statistical properties or of the quantum evolution of fermion systems requires the introduction of elements of an infinite dimensional Grassmann algebra and the integration over Grassmannian paths. For example, to describe a system with $$N$$ available quantum states, one introduces the generators $$\theta_i(\tau),$$ $$\bar\theta_i(\tau),$$ $$i=1\ldots N\ ,$$ of a Grassmann algebra. They satisfy the commutation relations $\theta_i(\tau)\theta_j(\tau')+\theta_j(\tau') \theta_i(\tau)=0\,,\quad \theta_i(\tau)\bar\theta_j(\tau')+\bar\theta_j(\tau') \theta_i(\tau)=0\,,\quad\bar\theta_i(\tau)\bar\theta_j(\tau')+\bar\theta_j(\tau') \bar\theta_i(\tau)=0\,.$ Then, rules of Grassmannian differentiation and integration can be formulated. It follows that the elements of the density matrix at thermal equilibrium, or the imaginary-time path integral, take the form $\left\langle \boldsymbol{\theta} '' | U (t'' ,t' ) | \bar {\boldsymbol{\theta}}' \right\rangle = \int^{ \boldsymbol{\theta} (t'' )= \boldsymbol{\theta} ''}_{\bar {\boldsymbol{\theta}} (t' )=\bar {\boldsymbol{\theta}}'} \left[ \mathrm{d} \boldsymbol{\theta} (\tau)\mathrm{d}\bar {\boldsymbol{\theta}} (\tau) \right] \mathrm{e}^{- \bar{\boldsymbol{\theta}} (t')\cdot\boldsymbol{\theta} (t')} \exp\left[-\mathcal{S} ( \boldsymbol{\theta} ,\bar {\boldsymbol{\theta}} )\right]$ with $\mathcal{S} ( \boldsymbol{\theta} , \bar{\boldsymbol{\theta}} ) = \int^{t''}_{t'} \mathrm{d} \tau\, \left\{ \bar{\boldsymbol{\theta}} (\tau)\cdot \dot {\boldsymbol{\theta}} (\tau)+H\left[ \boldsymbol{\theta} (\tau), \bar{\boldsymbol{\theta}} (\tau) \right] \right\} ,$ where $$H\left[ \boldsymbol{\theta} (\tau), \bar{\boldsymbol{\theta}} (\tau) \right]$$ represents the Hamiltonian acting on Grassmann functions.

## A generalization: The field integral

While the path integral is an interesting topic for its own sake, the most useful physics applications are provided by a generalization: the field integral, where the integration over paths is replaced by an integration over fields. For example, in a local field theory for a neutral scalar field $$\phi(x),$$ $$x\in\mathbb{R}^d\ ,$$ the partition function is given by $\mathcal{Z}=\int[\mathrm{d}\phi(x)]\mathrm{e}^{-\mathcal{S}(\phi)/\hbar},$ where the Euclidean action $$\mathcal{S}(\phi)$$ is a $$d$$-dimensional integral of a function of the field and its derivatives ($$\partial_\mu\equiv \partial/\partial x_\mu$$): $\mathcal{S}(\phi)=\int\mathrm{d}^d x\,\mathcal{L}_{\mathrm{E}}(\partial_\mu\phi(x),\phi(x)).$ While the algebraic properties of the path integral generalize easily, the field integral leads to new problems requiring new concepts like regularization and renormalization. Indeed, a natural physical choice, for example, would be $\tag{43} \mathcal{L}_{\mathrm{E}}(\partial_\mu\phi,\phi)=\frac{1}{2}\sum_\mu(\partial_\mu\phi)^2+ \frac{1}{2}r\phi^2+{g \over 4!}\phi^4,$

where $$r$$ and $$g \ge 0$$ are two parameters characterizing the model. However, in dimensions $$d>1$$ the derivative term no longer selects fields regular enough, as a discrete or lattice approximation reveals, and, as a consequence, field correlation functions are not defined at coinciding points. It is necessary to modify (in an unphysical way from the viewpoint of quantum physics) the action at short distance, a procedure called regularization. One possibility is to introduce quadratic terms in the field with derivatives of higher order $$2n>d\ ,$$ which restrict the integration to fields satisfying a Hölder condition as in $$d=1\ .$$ Another possibility is to consider a lattice approximation with a lattice spacing $$1/\Lambda\ .$$ The existence of a continuum limit or large $$\Lambda$$ limit, then requires, in addition, tuning the initial parameters of the model as a function of $$\Lambda\ ,$$ a procedure called renormalization. Renormalization, and its consequence, the renormalization group, find a natural interpretation in the theory of continuous macroscopic phase transitions.

In a relativistic-covariant quantum field theory, the real-time evolution (in 3+1 space-time dimensions) is then given by $\int[\mathrm{d}\phi(x)]\exp\left[{i\over\hbar}\mathcal{A}(\phi)\right],$ where $$\mathcal{A}$$ now is the classical action, space-time integral of the classical Lagrangian density $\mathcal{A}(\phi)=\int\mathrm{d}^4 x\,\mathcal{L}(\partial_\mu\phi(x),\phi(x)).$ In the example of the Euclidean Lagrangian (43), the real-time Lagrangian reads ($$t\equiv x_0$$) $\mathcal{L}=\frac{1}{2}(\partial_t \phi)^2-\frac{1}{2}(\nabla\phi)^2 -\frac{1}{2}r\phi^2-{g \over 4!}\phi^4,$ where $$\nabla\equiv \{\partial/\partial x^1,\partial/\partial x^2,\partial/\partial x^3\}\ .$$

Beside scalar boson fields, in general other types of fields are also required like Grassmann fields with spin for fermion matter. Moreover, since in the Standard Model that describes fundamental interactions at the microscopic scale, interactions are generated by the principle of gauge invariance, gauge fields also appear (and unphysical spinless fermions after quantization).