37.4 Interacting quantum fields

37.4 Interacting quantum fields

To describe an arbitrary number of particles moving in an external potential $V(x)$ , the Hamiltonian can be taken to be

$$\hat{H}={\int }_{-\infty }^{\infty }{\hat{\Psi }}^{†}(x)\left(-\frac{1}{2m}\frac{d^2}{d{x}^2}+V(x)\right)\hat{\Psi }(x)dx$$

If a complete set of orthonormal solutions ${\psi }_n(x)$ to the Schr¨odinger equation with potential can be found, they can be used to describe this quantum system using similar techniques to those for the free particle, taking as basis for ${\mathcal{H}}_1$ ₁ the ${\psi }_n(x)$ instead of plane waves of momentum $p.$ . A creation-annihilation operator pair $a_n,a_n^{†}$ is associated to each eigenfunction, and quantum fields are defined by

$$\hat{\Psi }(x)=\sum _n{\overline{\psi }}_n(x)a_n,{\hat{\Psi }}^{†}(x)=\sum _n{\psi }_n(x)a_n^{†}$$

For Hamiltonians quadratic in the quantum fields, quantum field theories are relatively tractable objects. They are in some sense decoupled quantum oscillator systems, although with an infinite number of degrees of freedom. Higher order terms in the Hamiltonian are what makes quantum field theory a dificult and complicated subject, one that requires a year-long graduate level course to master basic computational techniques, and one that to this day resists mathematician’s attempts to prove that many examples of such theories have even the basic expected properties. In the quantum theory of charged particles interacting with an electromagnetic field (see chapter 45), when the electromagnetic field is treated classically one still has a Hamiltonian quadratic in the field operators for the particles. But if the electromagnetic field is treated as a quantum system, it acquires its own field operators, and the Hamiltonian is no longer quadratic in the fields but instead gives an interacting quantum field theory.

Even if one restricts attention to the quantum fields describing one kind of particle, there may be interactions between particles that add terms to the Hamiltonian that will be higher order than quadratic. For instance, if there is an interaction between such particles described by an interaction energy $v(y-x)$ 2 this can be described by adding the following quartic term to the Hamiltonian

$$\frac{1}{2}{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{\hat{\Psi }}^{†}(x){\hat{\Psi }}^{†}(y)v(y-x)\hat{\Psi }(y)\hat{\Psi }(x)dxdy$$

The study of “many-body” quantum systems with interactions of this kind is a major topic in condensed matter physics.

Digression (The Lagrangian density and the path integral). While we have worked purely in the Hamiltonian formalism, another approach would have been to start with an action for this system and use Lagrangian methods. An action that will give the Schr¨odinger equation as an Euler-Lagrange equation is

$$\begin{array}{rl}S& ={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }\left(i\psi \frac{\partial }{\partial t}\overline{\psi }-h\right)dxdt\\ & ={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }\left(i\psi \frac{\partial }{\partial t}\overline{\psi }+\psi \frac{1}{2m}\frac{{\partial }^2}{\partial x^2}\overline{\psi }\right)dxdt\\ & ={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }\left(i\psi \frac{\partial }{\partial t}\overline{\psi }-\frac{1}{2m}|\frac{\partial }{\partial x}\psi {|}^2\right)dxdt\end{array}$$

where the last form comes by using integration by parts to get an alternate form of h as mentioned in section 37.2. In the Lagrangian approach to field theory, the action is an integral over space and time of a Lagrangian density, which in this case is 2 2

$$L(x,t)=i\psi \frac{\partial }{\partial t}\overline{\psi }-\frac{1}{2m}|\frac{\partial }{\partial x}\psi {|}^2$$

Defining a canonical conjugate momentum for $\overline{\psi }$ as $\frac{\partial L}{\partial \dot{\overline{\psi }}}$ gives as momentum variable $i\psi$. This justifies the Poisson bracket relation

$$\{\overline{\Psi }(x),i\Psi (x^{\prime })\}=\delta (x-x^{\prime })$$

but, as expected for a case where the equation of motion is first-order in time, the canonical momentum coordinate $i\Psi (x)$ is not independent of the coordinate $\overline{\Psi }(x)$ . The space ${\mathcal{H}}_1$ of wavefunctions is already a phase space rather than just a configuration space, and one does not need to introduce new momentum variables. One could try and quantize this system by path integral methods, for instance computing the propagator by doing the integral

$$\int D\psi (x,t)e^{\frac{i}{\hslash }S[\psi ]}$$

over paths in ${\mathcal{H}}_1$ parametrized by $t$, taking values from $t=0$ to $t=T$ . This is a highly infinite dimensional integral, over paths in an infinite dimensional space. In addition, recall the warnings given in chapter 35 about the problematic nature of path integrals over paths in a phase space, which is the case here.

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