Quantum field theory 1, lecture 05

5 Quantum Fields and Functional Integral

In this lecture we will start from many body quantum mechanics and construct the functional integral for a quantum field theory. In the last lecture we have shown how the operator formalism emerges from a functional integral, in short: functional integral \(\to \) operators. In this lecture we will proceed in the opposite direction. Starting from a formulation of many body quantum mechanics in terms of operators we will derive the equivalent functional integral, in short: operators \(\to \) functional integral. The aim of the lecture is once more to show the equivalence of the functional integral and the operator formalism. Historically, this is the way how Feynman introduced the functional integral for quantum mechanics. This construction of the functional integral can be found in many textbooks on quantum field theory at a somewhat later stage. The present lecture should also help to establish this contact.

In the present lecture we introduce quantum fields, establishing in this way the basic concepts of quantum field theory in the operator formalism. We construct the functional integral for quantum fields. We take the non-relativistic example of phonons. This demonstrates that quantum field theory is not only needed for relativistic particle physics. Phonons are perhaps also easier to understand intuitively than photons. There is not much conceptual difference between phonons and photons. Phonons are excitations in a solid, photons are excitations of the vacuum. Photons are relativistic.

5.1 Phonons as quantum fields in one dimension

One-dimensional crystal. Consider a one-dimensional crystal of atoms with lattice sites \( x_j = j \varepsilon \) and lattice distance \(\varepsilon \). Denote the displacement from the equilibrium position at \(x_j\) by \(Q_j\) and the momentum of the atoms by \(P_j\). The Hamiltonian for small displacements can be taken quadratic in \(Q_j\), and we decompose \(H = H_0 + H_\text{nn}\) with \begin{equation*} H_0 = \sum _j \left ( \frac{P_j^2}{2M}+\frac{D}{2}{Q_j}^2\right ), \quad \quad \quad H_\text{nn} = - \frac{B}{2} \sum _j Q_{j+1} Q_j. \end{equation*} Here \(Q_j\) and \(P_j\) are quantum operators with the usual commutation relations \begin{equation*} [Q_j, P_k] = i\delta _{jk}, \quad \quad \quad [Q_j, Q_k] = 0,\quad \quad \quad [P_i, P_j]= 0. \end{equation*} We use units where \(\hbar = 1\).

The term \(H_0\) alone describes decoupled harmonic oscillators at every lattice site \(j\). The term \(H_{nn}\) couples the oscillators by a next neighbour interaction. Phonons are thus described by a coupled system of harmonic oscillators.

Quantum fields. The displacements are an example for a quantum field, \begin{equation*} Q_j =Q(x). \end{equation*} Here \(x\) is a discrete variable labelling the lattice sites. In the continuum limit \(x\) will become a continuous position variable. The field \(Q(x)\) is an operator field. For each \(x\) one has an operator \(Q(x)\). Often such operator fields are called ”quantum fields”. We use this expression here as well, but not exclusively for the operator fields in the operator formalism. We will also employ the notion of quantum fields in the equivalent functional integral formalism that does not employ operators for its formulation. Also the momentum field \(P(x) = P_j\) is an operator field or quantum field. One may consider the pairs \(\{Q_j,P_j\}\) as a common (two-component) quantum field.

Occupation number basis. At each site \(j\) we define annihilation and creation operators \(a_j\) and \(a_j^\dagger \). The annihilation operators are \begin{equation*} a_j = \frac{1}{\sqrt{2}}\left ((DM)^{\frac{1}{4}} Q_j +i(DM)^{-\frac{1}{4}} P_j \right ), \end{equation*} and the creation operators are given by \begin{equation*} a_j^\dagger = \frac{1}{\sqrt{2}}\left ((DM)^{\frac{1}{4}} Q_j -i(DM)^{-\frac{1}{4}} P_j \right ). \end{equation*} The creation operators are the hermitian conjugates of the annihilation operators, \(a_j^\dagger = (a_j)^\dagger .\) The commutation relations are \begin{equation*} [a_j,a_k^\dagger ]= \delta _{jk}, \quad \quad \quad [a_j,a_k] = 0, \quad \quad \quad [a_j^\dagger ,a_k^\dagger ] = 0. \end{equation*} This can be verified by employing the commutation relations for \(Q\) and \(P\). Both \(a(x) = a_j\) and \(a^\dagger (x) = a_j^\dagger \) are operator fields.

Inserting \begin{equation*} Q(x) = Q_j =\frac{1}{\sqrt{2}}(DM)^{-\frac{1}{4}} \left (a_j +a_j^\dagger \right ), \end{equation*} and similar for \(P_j\), we can express the Hamiltonian in terms of \(a\) and \(a^\dagger \), \begin{equation*} H_0 = \omega _0 \sum _j \left (a_j^\dagger a_j +\frac{1}{2} \right ) = \omega _0 \sum _j \left (\hat{n}_j + \frac{1}{2} \right ), \end{equation*} with the frequency \( \omega _0 =\sqrt{D/M}\). You recognise the standard treatment of harmonic oscillators in quantum mechanics. Occupation numbers at positions \(x_j\) are expressed in terms of the operator \(\hat n_j = a_j^\dagger a_j\). They have the eigenvalues \(n_j = (0,1,2,\ldots )\). At each site \(j\) there are a number \(n_j\) of “localised phonons”. For \(B=0\) the system describes uncoupled harmonic oscillators, one at each lattice site.

We next discuss the effects of the next-neighbour interaction. It involves products of \(a_j\), \(a_{j+1}\) etc., according to \begin{equation*} \begin{split} H_\text{nn} &= -\frac{B}{2} \sum _j Q_{j+1} Q_j\\ &=-\frac{B}{2} \frac{(DM)^{-\frac{1}{2}}}{2}\sum _j \left (a_{j+1} + a^\dagger _{j+1} \right ) \left (a_j+a^\dagger _j \right ). \end{split} \end{equation*}

Momentum Space. It is possible to diagonalize \(H\) by a discrete Fourier transform. To this end, we write \begin{equation*} a_j = \frac{1}{\sqrt{\mathscr{N}}}\sum _q e^{i\varepsilon q j}a_q, \quad \quad \quad a_j^\dagger = \frac{1}{\sqrt{\mathscr{N}}} \sum _q e^{-i\varepsilon q j}a_q^\dagger . \end{equation*} Due to the finite lattice distance the sum is periodic in \(q\), \begin{equation*} \sum _q = \sum _{|q|\leq \frac{\pi }{\varepsilon }}, \end{equation*} and \(\mathscr{N} = \sum _j\) is a normalization factor corresponding to the number of lattice sites. If we place the sites of the lattice on a torus with circumference \(L\), the momentum sum is a discrete sum, with level distance given by \(2\pi /L\). If you are not familiar with these formulae you may look up in some text book a chapter on discrete Fourier transforms. It is the most simple and basic case for a lattice in solids.

Hamiltonian. We next express the Hamiltonian in terms of the Fourier modes. Insertion of \begin{equation*} \begin{split} Q_j &= \frac{1}{\sqrt{2\mathscr{N}}}(DM)^{-\frac{1}{4}}\sum _q (e^{i\varepsilon q j}a_q +e^{-i \varepsilon q j}a_q^\dagger )\\ &= \frac{1}{\sqrt{2\mathscr{N}}}(DM)^{-\frac{1}{4}}\sum _q e^{i\varepsilon q j}\left (a_q +{a_{-q}}^\dagger \right ), \end{split} \end{equation*} yields \begin{equation*} H_\text{nn} = -\frac{B}{4\mathscr{N}} (DM)^{-\frac{1}{2}}\sum _j\sum _q\sum _{q'} e^{i\varepsilon q' j} e^{i \varepsilon q(j+1)} \left (a_q +{a_{-q}}^\dagger \right )\left (a_q' +{a_{-q'}}^\dagger \right ). \end{equation*} We use the following identity for discrete Fourier transforms, \begin{equation*} \sum _j e^{i\varepsilon (q+q')j} = \mathscr{N} \delta _{q,-q'}, \end{equation*} which corresponds to the familiar continuum expression \begin{equation*} \int dx \, e^{i(q+q')x} = 2\pi \delta (q+q^\prime ). \end{equation*} One obtains \begin{equation*} \begin{split} H_\text{nn} &= -b\sum _q e^{i\varepsilon q }\left (a_q +{a_{-q}}^\dagger \right ) \left (a_{-q}+a_q^\dagger \right )\\ &= -b \sum _q \cos (\varepsilon q) \left (a_q + a_q^\dagger \right ){\big (}a_{-q}+a_{-q}^\dagger{\big )}, \end{split} \end{equation*} with \(b = \frac{B}{4}(DM)^{-\frac{1}{2}}\). Similarly, one has \begin{equation*} H_0 = \omega _0 \sum _q\left (a_q^\dagger a_q +\frac{1}{2}\right ). \end{equation*}

Momentum modes. At this stage, the Hamiltonian \(H\) involves separate \(q\)-blocks, \begin{equation*} H = \sum _q H_q, \end{equation*} with \begin{equation*} H_q = \omega _0 \left (a_q^\dagger a_q +\frac{1}{2}\right ) -b \cos (\varepsilon q){\Big (}a_q +{a_{-q}}^\dagger{\Big )}{\Big (}a_{-q} + a_q^\dagger{\Big )}. \end{equation*} Each block involves \(q\) and \(-q\). What remains is the diagonalization of the \(q\)-blocks, done by the Bogoliubov transformation, \begin{equation*} a_q = \alpha (q) A_q + \beta (q) A_{-q}^\dagger ,\quad \quad \quad a_q^\dagger = \alpha (q) A_q^\dagger + \beta (q) A_{-q}, \end{equation*} where the commutation relations \begin{equation*} [a_q,a_q^\dagger ]=1,\quad \quad \quad [A_q, A_q^\dagger ] = 1, \end{equation*} require \begin{equation*} \alpha (q)^2 - \beta (q)^2 = 1. \end{equation*}

The coefficients \(\alpha (q)\) are determined such that the Hamiltonian is diagonal, \begin{equation*} H=\sum _q \omega _q \left (A_q^\dagger A_q + \frac{1}{2}\right ). \end{equation*} The algebra is straightforward and one finds for the squared frequencies of the independent oscillation modes \begin{equation*}{\omega _q}^2=\frac{D}{M}\left (1-\frac{B}{D} \cos (\varepsilon q) \right ). \end{equation*} In the momentum basis the phonons are described as uncoupled harmonic oscillators, one for every momentum \(q\). They are a free quantum field, which means that they do not interact with themselves.

Quantum field theory So far we have just presented the most basic notion for a quantum description of solids. Conceptually, this is simply a quantum theory for many degrees of freedom. Phonons are a simple example for a quantum field theory. No additional concepts need to be introduced. The so called ”second quantisation” is nothing else than quantum mechanics for many degrees of freedom. The continuum limit, for which \(x\) becomes a continuous variable, does not introduce any qualitative changes.

Many properties of quantum field theories, as the role of the vacuum and particles as excitations of the vacuum, can already be seen for phonons. The vacuum obeys, as usual \(A_q |0\rangle = 0\). This is not the same as for \(B=0\), where one has \(a_q| 0 \rangle = 0\). The vacuum state depends on \(B\). It can be a complicated object. For phonons it remains possible to construct the vacuum state explicitly. For more complicated quantum field theories this is, in general, no longer possible. Phonons are considered as excitations of the vacuum. These excitations are called quasiparticles or simply particles. Their properties depend on the vacuum, e. g. the dispersion relation depends on \(B\). This concept plays an important role for elementary particle physics. For example, the mass of the electron depends on the expectations value of the Higgs field in the vacuum state. An important insight may be phrased in the simple term: ”The vacuum is not nothing.”

Dispersion relation. The relation between frequency and momentum, \begin{equation*} \omega (q)= \omega _q = \sqrt{\frac{D-B \cos (\varepsilon q)}{M}}, \end{equation*} is called the dispersion relation. Consider the limit of small \(\varepsilon q\), where one can expand, \(\cos (\varepsilon q) = 1- \frac{1}{2}\varepsilon ^2 q^2\), such that \begin{equation*}{\omega ^2 (q)} = \frac{D-B}{M} + \frac{\varepsilon ^2 B}{2M}q^2. \end{equation*} In our units frequency and energy are identical, such that the dispersion relation corresponds to the energy momentum relation of the phonon-quasi-particles.

For \(D>B\) the occupation relation has a gap, one needs positive energy even for a phonon with \(q=0\). For many cases the interaction between atoms is of the form \( (Q_j - Q_{j-1})^2 \), involving only the distance between two neighbouring atoms. Then \(D=B\), phonons are gapless and the dispersion relation becomes linear for small \(\varepsilon q\). The sound velocity is given here by \begin{equation*} v=\left |{\frac{d \omega }{d q}} \right |= \frac{\varepsilon ^2 B q}{2M \omega (q)}. \end{equation*}

Generalisations. In three dimensions \(d=3\) one has \(q \to \vec q\) and the dispersion relation becomes an equation for \(\omega (\vec q)\). For real solids it depends on the particular structure of the lattice and the form of the interactions.

Continuum limit. The continuum limit can be taken for situations where the expectation values of the relevant observables and corresponding wave functions are sufficiently smooth. This means that their variation with \(x\) is small on scales of the order \(\varepsilon \). Typically, this concerns properties dominated by modes with low momenta \(q\). The continuum limit corresponds to the limit \(\varepsilon q\to 0\).

Photons. For photons the dispersion relation is (in units where the velocity of light is unity, \(c=1\)), \begin{equation*} \omega (\vec q) = |\vec q|. \end{equation*} There are two photon helicities, related to polarisation. Photons are conceptually very similar to phonons. We will discuss them in more detail later.

Quantum fields for photons. For photons, associated quantum fields are the electric field \(\vec E(\vec q)\) in momentum space or \(\vec E(\vec x)\) in position space, as well as the magnetic field \(\vec B(\vec q)\) or \(\vec B(\vec x)\), respectively. In other words, the electric field \(\vec E\) and the magnetic field \(\vec B\) are quantum operators! The corresponding operator fields consist of operators for each \(\vec x\) or for each \(\vec q\). There is conceptually no difference to phonons.

Bosonic atoms without interaction. For free, non-relativistic atoms, the dispersion relation is given by \begin{equation*} \omega (\vec q) = \frac{\vec q^2}{2M}. \end{equation*} For the grand-canonical ensemble, one includes a chemical potential, multiplying the total particle number. This shifts effectively \begin{equation*} \omega (\vec q) \to \epsilon (\vec q) = \frac{\vec q^2}{2M}-\mu . \end{equation*} We will not distinguish \(\omega (\vec q)\) and \(\varepsilon (\vec q)\) unless stated otherwise.

General free quantum field theories. Formulated in momentum space, free quantum field theories are described by separate harmonic oscillators for each momentum mode \(q\). The detailed microscopic origin of the Hamiltonian does not matter. All properties are encoded in the particular form of \(H_q\), as the dispersion relation. Phonons, photons or bosonic atoms have all the same status. This extends to excitations or quasiparticles in many domains of physics.

5.2 Functional integral for quantum fields

In this part we introduce the functional integral for quantum fields. We discuss both thermodynamic equilibrium and the time evolution for given initial conditions. The mathematical formalism is very similar for both cases. They are distinguished by an important factor of (i) multiplying the action. While this is crucial for the physical behaviour, the mathematical treatment for both cases is identical. We can construct the functional integral simultaneously for the equilibrium situation and for quantum dynamics.

Free quantum boson gas in thermal equilibrium. We start with quantum statistics for free fields. Quantum statistics is distiguished from the classical statistics discussed in the previous lecture by the operator nature of the quantum fields. We will, nevertheless, derive a functional integral formulation involving only commuting objects. This formulation involves one additional dimension of ”euclidean time”.

For the Hamiltonian \begin{equation*} H = \sum _q \omega (q) \left (a_q^\dagger a_q +\frac{1}{2}\right ), \end{equation*} the partition function in thermal equilibrium is given by the trace \begin{equation*} Z= \text{Tr} \, e^{-\beta H}, \end{equation*} with \(\beta = \frac{1}{k_\text{B} T}= \frac{1}{T}\). (We use units for the Boltzmannn constant \(k_\text{B}=1\)). It decays into independent factors for every momentum mode, \begin{equation*} Z= \prod _q \text{Tr} \, e^{-\beta \omega _q \left (a_q^\dagger a_q + \frac{1}{2}\right )} = \prod _q Z_q. \end{equation*} One only has to compute the individual \(Z_q\), \begin{equation*} Z = \text{Tr} \, e^{-\tilde \beta \left (a^\dagger a + \frac{1}{2}\right )}, \end{equation*} with \(\tilde \beta = \beta \omega _q\) (we omit the index \(q\)). As an example, for a free gas of bosonic atoms one has \begin{equation*} \omega (q) = \frac{\vec q^2}{2M}- \mu , \end{equation*} with chemical potential \(\mu \). The logarithm of the partition function is simply a momentum sum of the individual logarithms. From the logarithm of \(Z(\beta ,\mu )\) one can derive all thermodynamics of the quantum boson gas. This will be done in lecture 6 including interactions.

In this lecture we will derive a functional integral representation of the partition function \begin{equation*} Z = \text{Tr} \, e^{-\beta H} = \int D\phi \; e^{-S[\phi ]}, \end{equation*} with Euclidean action \begin{equation*} S = \int _{-\frac{\beta }{2}}^{\frac{\beta }{2}} d\tau \sum _q \phi ^* (\tau ,q) \left (\frac{\partial }{\partial \tau }+ \omega (q) \right ) \phi (\tau ,q). \end{equation*} The complex fields \(\phi (\tau ,q)\) are periodic, \begin{equation*} \phi (\tau +\beta ,q)=\phi (\tau ,q). \end{equation*} In consequence, the euclidean time \(\tau \) parameterizes a torus with circumference \(\beta \).

Partition function with boundary conditions. We will derive the functional integral below. In order to do this in parallel for the dynamical evolution in quantum field theory we introduce a formal boundary term in the expression \begin{equation*} \tilde Z = \text{Tr}\left \{ b \, e^{-\tilde \beta \left (a^{\dagger }a + \frac{1}{2} \right )} \right \}. \end{equation*} For \(b=1\) one has \(\tilde Z = Z\) for thermodynamic equilibrium if \(\tilde \beta = \beta \omega \) is real. A more general boundary term \(b\) has no direct physical meaning for the thermal equilibrium state of phonons, photons or atoms. It is used here as a technical device which permits us to discuss the functional integral for a larger class of operator problems. The boundary term \(b\) is a matrix in Hilbert space. For example, in the occupation number basis one has \begin{equation*} \tilde Z = b_{nm}\left (e^{-\tilde \beta \left (a^\dagger a + \frac{1}{2}\right )} \right )_{mn}. \end{equation*} We may take the “boundary term” \(b\) as a product of wave functions, \begin{equation*} b_{nm} = (\psi _\text{in})_n (\phi _\text{f})_m, \end{equation*} such that \begin{equation*} \begin{split} \tilde Z &= (\phi _\text{f})_m \left (e^{-\tilde \beta \left (a^\dagger a + \frac{1}{2}\right )}\right )_{mn}(\psi _\text{in})_n \\ &= \left \langle \phi _\text{f} \left | e^{-\tilde \beta \left (a^\dagger a +\frac{1}{2} \right )} \right | \psi _\text{in} \right \rangle . \end{split} \end{equation*}

Extension to complex formulation. The trace is well defined also for complex values of \(\tilde \beta \). In particular, we may consider purely imaginary \(\tilde \beta \), \begin{equation*} \tilde{\beta } = i\omega \Delta t. \end{equation*} We can also choose a complex boundary term \(b\) and admit complex wave functions \(\phi _\text{f}\) and \(\psi _{\text{in}}\). We employ the notation of quantum mechanics with \(\langle \phi _\text{f} | \) involving complex conjugation, e. g. \( \langle \phi _\text{f} |_m = (\phi _\text{f}^*)_m. \) In general, \(\tilde Z \) will now be a complex number.

Transition amplitude. With this setting \(\tilde Z\) is the transition amplitude for the quantum mechanics of an harmonic oscillator, \begin{equation*} \begin{split} \tilde Z &= \langle \phi _\text{f} | e^{-i\Delta t \omega \left (a^\dagger a + \frac{1}{2}\right )} | \psi _\text{in}\rangle \\ &= \langle \phi _\text{f} | e^{-i \Delta t H} | \psi _\text{in}\rangle . \end{split} \end{equation*} The operator \(e^{-i \Delta t H}\) is the evolution operator in quantum mechanics between an initial time \(t_{in}\) and a final time \(t_f=t_{in}+\Delta t\). We associate the boundary wave functions with \begin{equation*} \psi _\text{in} = \psi (t_{\text{in}}),\quad \quad \quad \phi _f = \phi (t_\text{f}), \end{equation*} In quantum mechanics the evolution operator relates the wave function at \(t_f\) to the initial wave function at \(t_{in}\) \begin{equation*} \psi (t_\text{f}) = e^{-i(t_\text{f}-t_\text{in}) H} \psi (t_\text{in}). \end{equation*} We can therefore also interpret the quantity \(\tilde Z\) as the transition amplitude between \(\psi \) and \(\phi \) at the common time \(t_\text{f}\), \begin{equation*} \tilde Z = \langle \phi (t_\text{f})| \psi (t_\text{f})\rangle , \quad \quad \quad \Delta t = t_\text{f}-t_{\text{in}}. \end{equation*} The square \(|\tilde Z|^2\) measures the probability that a given \(\psi (t_\text{in})\) coincides at \(t_\text{f}\) with \(\phi (t_\text{f})\).

We can generalise the single harmonic oscillator to a free quantum field theory. The Hamiltonian is a sum over Hamiltonians for every momentum mode \(q\). Then \(H = \omega \left (a^\dagger a + \frac{1}{2} \right )\) stands for \(H_q\). With total Hamiltonian being the sum of all \(H_q\), the expression \(\tilde Z\) is the transition amplitude for a free quantum field theory. Adding interactions the transition amplitude is a key element for the S-matrix for scattering to be discussed in coming lectures.

Split into factors. The trace can be evaluated by splitting \(\tilde \beta \) into small pieces, and therefore \(e^{-\tilde \beta H}\) into many factors. For the transition amplitude this factorizes the evolution operator into many evolution operators for small time steps. For thermal equilibrium there is no such intuitive interpretation for small steps in euclidean time. Nevertheless, the method of splitting \(\Delta t\) or \(\tilde \beta \) into small steps is the same.

We demonstrate this method for a single harmonic oscillator. The split of \(\tilde \beta \) into small steps is done by writing \(\tilde \beta = (2N+1)\delta \), where \(|\delta | \ll 1\). For convenience we assume \(N\) to be even. The factorization yields \begin{equation} \exp \left \{-\tilde \beta \left [a^\dagger a +\frac{1}{2} \right ] \right \} = \prod _{j=-N}^N \exp \left \{ -\delta \left [a^\dagger a + \frac{1}{2}\right ]\right \}. \label{eq:exponentialSplit} \end{equation} The splitting is a formal method and the index \(j\) has nothing to do with lattice sites or other physical objects. For large \(N\) or small \(\delta \), the exponential simplifies. This would not be necessary for the present very simple case, but is very useful for more complicated Hamiltonians which involve pieces that do not commute with each other.

The split will be used to define a functional integral. Indeed, the expression \eqref{eq:exponentialSplit} looks already like a product of transfer matrices. We can take \(N \to \infty \) such that approximations for small \(\delta \) become exact. Let us define the operators \begin{equation*} \hat{x} = \frac{1}{\sqrt{2}}\left (a^\dagger +a\right ),\quad \quad \quad \hat{p} = \frac{i}{\sqrt{2}}\left (a^\dagger -a \right ), \end{equation*} with commutation relation \begin{equation*} [\hat{x},\hat{p}] = i. \end{equation*} The operators \(\hat{x}\) and \(\hat{p}\) have similar properties as position and momentum operators. In our context they are abstract operators, since for photons or phonons already \(a^\dagger a\) stands for \(a^\dagger _q a_q\) or \(A^\dagger _q A_q\) in momentum space. Thus \(\hat{x}\) and \(\hat{p}\) have nothing to do with position and momentum of phonons or photons.

In terms of the operators \(\hat{x}\), \(\hat{p}\) one has \begin{equation*} \hat{H} = a^\dagger a +\frac{1}{2} = \frac{\hat{p}^2}{2} + V(\hat{x}),\quad \quad \quad V(\hat{x}) = \frac{\hat{x}^2}{2}. \end{equation*} This yields the expression \begin{equation*} \exp{\left \{-\tilde{\beta }\left [a^\dagger a +\frac{1}{2}\right ]\right \}} = \prod _{j=-N}^N \exp \left \{-\delta \left [\frac{\hat{p}^2}{2} + V(\hat{x})\right ] \right \}, \end{equation*} where \begin{equation*} \tilde H = \frac{\hat{p}^2}{2} + V(\hat{x}). \end{equation*} For a general function \(V(\hat{x})\) this is the Hamiltonian for one-dimensional quantum mechanics in a potential \(V\), with a factor \(1/M\) incorporated in \(\delta \). Many steps below are valid for general \(V\). Our treatment covers the path integral for a quantum particle in a potential.

Eigenfunctions of \(\hat{x}\) and \(\hat{p}\). We define eigenfunctions of the operators \(\hat{x}\) and \(\hat{p}\), \begin{equation*} |x\rangle \quad \quad \quad \text{such that} \quad \quad \quad \hat{x}|x\rangle = x |x\rangle , \end{equation*} and \begin{equation*} |p\rangle \quad \quad \quad \text{such that} \quad \quad \quad \hat{p}|p\rangle = p|p\rangle . \end{equation*} Here \(x\) and \(p\) are continuous variables. We can choose a normalization such that \begin{equation*} \langle x' | x \rangle = \delta (x'-x), \quad \quad \quad \langle p' | p \rangle = 2\pi \delta (p'-p), \end{equation*} and \begin{equation*} \int dx \, |x\rangle \langle x| = \mathbb{1}, \quad \quad \quad \int \frac{dp}{2\pi } \, |p\rangle \langle p| = \mathbb{1}. \end{equation*} We insert complete systems of eigenfunctions between each of the factors, \begin{equation*} \prod _{j={-N}}^N e^{-\delta \tilde H} = \left [\prod _{j=-N}^{N+1} dx_j \right ] |x_{N+1}\rangle \langle x_{N+1}|e^{-\delta \tilde H}|x_N\rangle \langle x_N|\cdots |x_{1-N}\rangle \langle x_{1-N}| e^{-\delta \tilde H}|x_{-N}\rangle \langle x_{-N}|. \end{equation*}

Evaluation of factors. The factors \(\langle x_{j+1}|e^{-\delta \tilde{H}}|x_j\rangle \) are complex numbers, no longer operators. For their computation it is convenient to insert a complete set of \(\hat{p}\) -eigenstates, \begin{equation*} \langle x_{j+1}| e^{-\delta \tilde{H}}|x_j\rangle = \int \frac{dp_j}{2\pi }\langle x_{j+1}| p_j \rangle \langle p_j| e^{-\delta \tilde{H}}|x_j\rangle . \end{equation*} We next use for \(\delta \to 0\) the expansion \begin{equation*} \exp \left \{-\delta \left [\tfrac{\hat{p}^2}{2}+V(\hat{x})\right ]\right \} = \exp \left \{-\delta \tfrac{\hat{p}^2}{2}\right \} \exp \left \{- \delta V(\hat{x}) \right \} +{\cal O}(\delta ^2), \end{equation*} where the term \(\sim{\cal O}(\delta ^2)\) arises from the commutator of \(\hat{x}\) and \(\hat{p}.\) Corrections \(\sim \) \(\delta ^2\) can be neglected for \(\delta \to 0\) such that \begin{equation*} \langle x_{j+1}| e^{-\delta \tilde{H}}|x_j\rangle = \int \frac{dp_j}{2\pi }e^{-\delta \frac{p_j^2}{2}}e^{-\delta V(x_j)}\langle x_{j+1}|p_j\rangle \langle p_j|x_j\rangle . \end{equation*} No operators appear anymore in this expression and we only need \begin{equation*} \langle p_j| x_j\rangle = e^{-i p_j x_j}, \quad \quad \quad \langle x_{j+1}|p_j\rangle \langle p_j|x_j\rangle = e^{ip_j(x_{j+1}-x_j)}. \end{equation*} This yields the expression \begin{equation*} \langle x_{j+1}|e^{-\delta \tilde{H}}|x_j\rangle = \int \frac{dp_j}{2\pi } \exp \left \{ ip_j(x_{j+1}-x_j) - \delta \left [ \tfrac{p_j^2}{2}+V(x_j)\right ] \right \}. \end{equation*}

Functional integral. Insertion of these factors yields \begin{equation*} e^{-\tilde{\beta }\tilde{H}} = \int dx_{-N} \int dx_{N+1}|x_{N+1}\rangle \; F \; \langle x_{-N}|, \end{equation*} with \begin{equation*} F = \int D\phi ' \exp \left \{\sum _{j={-N}}^N\left [ip_j(x_{j+1}-x_j)-\delta \tfrac{p_j^2}{2}+\delta V(x_j) \right ]\right \}, \end{equation*} and functional measure \begin{equation*} \int D\phi '= \left [\prod _{j=-N+1}^N \int _{-\infty }^\infty dx_j \right ] \left [\prod _{j= -N}^N \int _{-\infty }^\infty \frac{dp_j}{2\pi }\right ]. \end{equation*} With boundary terms one obtains \begin{equation*} \langle \phi _\text{f}|e^{-\tilde{\beta }\tilde{H}}|\psi _{\text{in}}\rangle = \int dx_{-N}\int dx_{N+1}\langle \phi _f|x_{N+1}\rangle \; F \; \langle x_{-N}|\psi _{\text{in}}\rangle . \end{equation*}

Summary. In conclusion, we have transformed the operator trace into a functional integral \begin{equation*} \tilde Z = \text{Tr}\left \{ b \, e^{-\tilde \beta \left (a^{\dagger }a + \frac{1}{2} \right )} \right \} =\langle \phi _\text{f}|e^{-\tilde{\beta }\tilde{H}}|\psi _{\text{in}}\rangle =\int D\phi \, e^{-S}. \end{equation*} The action is given by \begin{equation*} S= -\sum _{j=N}^N \left \{ ip_j(x_{j+1}- x_j)- \delta \left [\tfrac{p_j^2}{2}+V(x_j) \right ]\right \}, \end{equation*} and the integration measure reads \begin{equation*} \int D\phi = \left [\prod _j \int dx_j \int \frac{dp_j}{2\pi }\right ]. \end{equation*} The boundary factor \(\tilde{b}\) has the form \begin{equation*} \tilde{b} = \int dx_{-N}\int dx_{N+1}\langle \phi _f|x_{N+1}\rangle \langle x_{-N}|\psi _{\text{in}}\rangle . \end{equation*}

From this expression Feynman’s path integral obtains by performing the Gaussian integration over the variables \(p_j\). What remains is an integral over all possible paths \begin{equation*} \int Dx[t] = \prod _j \int dx_j . \end{equation*} This is not the direction we follow in this lecture. We rather develop a formulation with complex variables. This can then easily be extended to a field theory.