The Yukawa Interaction in Ordinary Quantum Mechanics

Quantum field theory can be a challenging subject to learn and teach, but one approach towards quantum field theory pedagogy is to first focus on fields in one spacetime dimension ($d=1$). This has the advantage that it may be connected to ordinary quantum mechanical systems which may be solved in terms of methods which may already be familiar to a student, such as the use of wave functions. Certain features of quantum field theory calculations, such as the appearance of divergences and use of Grassmann numbers, may meet with some initial skepticism. So it may aid understanding to first introduce these concepts in a system which may be understood from an alternate, more familiar perspective. This also has the benefit of allowing a student to practice the new technical details of quantum field theory, such as the combinatorics of Feynman diagrams, on a problem which they may already be confident in solving a different way.

Unfortunately this article can not take such a broad scope as to teach the reader these basic technical details for the first time. Rather it is directed either towards teachers of quantum field theory who are looking for an enlightening example for their course, or for students who may have already read a textbook on quantum field theory (e.g. [1, 2, 3]) and have already been exposed to Feynman diagrams and path integrals, but are looking to solidify their understanding.

A similar approach towards introducing quantum field theory methods by considering fields in $d=1$ has already been considered by Boozer [4] who discussed both the $\phi^{4}$ interaction, which may be understood as an anharmonic oscillator in ordinary quantum mechanics, as well the Yukawa interaction, a system which involves fermions and which will be introduced more thoroughly below. A fundamental difference from Boozer’s approach is that we shall consider this from the point of view of path integrals in Euclidean time rather than the point of view of canonical quantization.

In particular, the treatment of fermions in terms of path integrals over Grassmann numbers may be something that can seem quite disconnected from previously learned methods in physics, and it may be beneficial to introduce it in terms of a system that may be easily solved in ordinary quantum mechanics. Of course in one spacetime dimension there is no non-trivial Lorentz group so fermion fields will lack a natural spinor index, but otherwise their propagator takes a very similar form to that in higher dimensional field theories. An interacting theory such as we will present here will involve expansions in terms of Feynman diagrams which will take essentially the exact same form as those in genuine quantum field theories in higher dimension. But from the perspective of canonical quantization, a single fermion field in $d=1$ is just associated to a ladder operator in a two-level system. So it is interesting to see how this simple two-level picture could possibly be compatible with all the complexity of fermion propagators in an asymptotic series of Feynman diagrams. Seeing how this works out explicitly is to a large extent the main goal of the present article.

Part of the motivation for considering $d=1$ fermions in more detail here comes from a related study of constrained quantum mechanical systems such as the quantum rotor from a quantum field theory perspective [5]. There it was pointed out that from the perspective of canonical quantization the fermionic corrections to the bosonic correlation functions of the supersymmetric quantum rotor should collectively vanish, even though this does not occur in higher spacetime dimension. So it was interesting to see what special features of fermions in $d=1$ lead to such a cancellation from a purely path integral perspective.

But to better understand fermionic fields in $d=1$, a far simpler model than a supersymmetric non-linear sigma model may be considered. Perhaps the simplest interacting theory involving a scalar field $\phi$ and fermionic field $\bar{\psi},\psi$, is just the Yukawa interaction which has the following action integrated over Euclidean time $\tau$,

This just has the usual quadratic terms for a scalar and fermion field extrapolated down to $d=1$, and an interaction term $\phi\bar{\psi}\psi$ which is the main characteristic of the model. The fields $\phi$ and $\psi$ have ‘bare masses’ $m$ and $\mu$ respectively. The derivative term for the fermion field may appear a little unusual but this is just due to the triviality of one-dimensional Clifford algebras and lack of spinor indices. If it is desired to emphasize the connection to higher dimensional theories, a single one component gamma ‘matrix’ $\gamma^{\tau}=i$ may be introduced in terms of which fermionic derivative term may also be expressed as $-i\bar{\psi}\gamma^{\tau}\partial_{\tau}\psi$. Note that we are setting $\hbar=1$ so that the action is dimensionless. Given that $\tau$ has dimensions of length, this implies that the the scalar field $\phi$ has dimensions of the square root of length, $\psi$ is dimensionless, and the coefficient $\lambda$ of the interaction term must have dimensions of length to the $-3/2$ power.

Now let us consider this system from the point of view of ordinary quantum mechanics. The connection between operator formalism and the Euclidean path integral formalism for $d=1$ is reviewed for instance in Chapter 1 of [3]. The action $S$ is associated to a a Hamiltonian $H$ expressed in terms of position operator $q$, momentum operator $p$, and fermionic creation and annihilation operators $c^{\dagger},c$,

Actually the ordering of the operators in the interaction term is not specified fully by the equal time $\phi\bar{\psi}\psi$ term in the path integral. We shall return to this issue later, but for now we are simply taking this ordering as given and will choose our regularization in the path integral approach in order to ensure this holds.

From this Hamiltonian we can see that the Yukawa interaction is actually extremely simple in $d=1$. The Hamiltonian commutes with the number operator $N\equiv c^{\dagger}c$, so we may consider the $N=0$ and $N=1$ eigenspaces separately. For states with $N=0$, which may be called the bosonic sector, the Hamiltonian reduces to an ordinary harmonic oscillator Hamiltonian with angular frequency $m$

which has spectrum $E_{n,B}=\left(n+\frac{1}{2}\right)m.$ For the fermionic sector with $N=1$, the Hamiltonian $H_{F}$ just reduces to a shifted harmonic oscillator by completing the square,

Here the constant shift in the Hamiltonian may be considered the energy $\mu_{\lambda}$ of the fermion itself

The spectrum in the fermion sector is just $E_{n,F}=E_{n,B}+\mu_{\lambda}$, so as far as the spectrum is concerned the Yukawa interaction only serves to correct the energy of the fermionic excitation from $\mu$ to $\mu_{\lambda}$. The interaction also leads to a non-zero expectation value for $q$, since the minimum of the harmonic oscillator potential is shifted,

Note that if $\lambda$ is large enough so that $\mu_{\lambda}$ becomes negative, then the true ground state is in the fermionic sector. From the quantum field theory perspective this sounds a bit more impressive. As $\lambda$ increases the system goes into a phase where the $\bar{\psi}\psi$ operator ‘condenses’ and the bosonic field $\phi$ picks up a vacuum expectation value. But from the quantum mechanical perspective this is all very straightforward.

It is now our goal to see how the simple quantum mechanical picture described above is consistent with the correlation functions calculated perturbatively from the path integral. The demonstration of this will be organized as follows. First in Sec. II we will very briefly review the non-perturbative connection between Euclidean correlation functions and the spectrum of the underlying quantum mechanics. In Sec. III we move on to calculating these correlation functions in the vacuum at zero temperature. First the technical details of regularization and its relation to operator ordering must be considered, but then the expressions for $\mu_{\lambda}$ and $\langle q\rangle_{F}$ above will be rederived through a Feynman diagram calculation.

One of peculiarities of fermions in $d=1$ from the path integral perspective will be shown to be the vanishing of all loops involving more than one fermion propagator. This may be rather puzzling at first because the perturbative expansion will involves all sorts of diagrams involving fermion loops, and it seems that these must be playing some role in the theory.

This issue will be clarified in Sec. IV by considering correlation functions at non-zero temperature where the fermion loops no longer vanish. The fermion pair correlation functions investigated here are familiar from many-body quantum mechanics (see e.g. [6]) and it is shown how a delicate cancelation takes place in order for the fermion field bilinear $\bar{\psi}\psi$ to be consistent with its quantum mechanical interpretation as a number operator in a simple two-level system. Finally an alternative summary of the results of this paper is given in Sec. V, and readers may freely skip ahead.

The path integral approach gives a natural way to calculate correlation functions of operators which are ordered in Euclidean time. This section will review how information about the quantum mechanical spectrum may be extracted from the two-point correlation functions. This is essentially a simpler version of the same idea that underlies the well-known Källén-Lehmann spectral representation in standard quantum field theory, which is treated for instance in Sec. 7.1 of [1]. This material is also covered in Boozer’s article [4], and the reader may find the related discussion of the quantum rotor in the first three sections of [5] helpful as it spends more time discussing the paradigm shift between quantum mechanics and quantum field theory and begins with zero-dimensional path integrals as a warm up case.

One advantage of considering Euclidean time $\tau$ rather than ordinary Minkowski time $t=-i\tau$ is due to the formal similarity between the Euclidean time translation operator $e^{-\tau H}$ and the Boltzmann factor involved in thermal expectation values. A thermal expectation value of an operator $\hat{O}$ is given by a trace over eigenstates $|n\rangle$ of the Hamiltonian.

In the zero temperature $\beta\rightarrow\infty$ limit this just becomes a vacuum expectation value, and that will be our first focus.

The Euclidean time ordered correlation functions in the bosonic ($N=0$) ground state $|0_{B}\rangle$ will be enough to tell us the energies of a single bosonic or fermionic excitation. To be clear, the Euclidean time dependence of an operator is given by $\hat{O}(\tau)=e^{H\tau}\hat{O}e^{-H\tau}$. Inserting an identity operator in the form $I=\sum_{n}|n\rangle\langle n|$ next to a Euclidean field allows us to replace the time translation operator by its eigenvalue acting on the state $|n\rangle$. Using this trick the time-ordered two-point correlation function for $q$ evaluates to

The Fourier transformed form in the second line may be verified by direct contour integration in the complex $p$ plane. If $\tau_{f}-\tau_{i}>0$ then we must close the integration contour in the negative imaginary half plane which encloses the pole $p=-i(E_{n}-E_{0})$, and vice versa if $\tau_{f}-\tau_{i}<0$. This correctly produces the absolute value in the exponential arising from the Euclidean time ordering.

So from this expression it is seen that the poles of the two-point function in momentum space give us information about the spectrum $E_{n}-E_{0}$. The simple system considered here (2) is essentially a harmonic oscillator and $q|0_{B}\rangle$ is exactly the first excited state up to normalization. So all the amplitudes $\left|\langle n|q|0_{B}\rangle\right|^{2}$ will vanish except for the level $n=1$. The difference $E_{1}-E_{0}$ is just the harmonic oscillator frequency $m$, and this is uncorrected even in the presence of non-zero $\lambda$.

The fermionic correlation function in the bosonic vacuum follows similarly from the same trick of inserting the identity operator expressed in terms of energy eigenstates

Here $\theta(\tau_{f}-\tau_{i})$ is a Heaviside function which vanishes if $\tau_{f}<\tau_{i}$ since due to the time ordering in that case the annihilation operator $c(\tau_{f})$ would appear to the right acting on the vacuum $|0_{B}\rangle$. As before it may be verified by direct contour integration that this is equal to

Once again the poles give information on the spectrum, but in this case the infinite sum over $n$ will remain since the harmonic oscillator eigenstates in the fermionic sector have shifted wavefunctions. As far as the $q$ and $p$ operators are concerned, the state $c^{\dagger}|0_{B}\rangle$ is a Gaussian centered at $q=0$ just like $|0_{B}\rangle$ itself, so it has non-zero overlap with the excited harmonic oscillator states which are centered at $q=\langle q\rangle_{F}$ defined in (6). But to lowest order in $\lambda$, $c^{\dagger}|0_{B}\rangle$ only has overlap with the fermionic ground state $|0_{F}\rangle$, and that has an energy difference $E_{0,F}-E_{0,B}$ which is equal to $\mu_{\lambda}$ in (5).

We wish to compare these correlation functions to those found via the path integral approach using action (1). A good introduction to Euclidean path integrals including the finite $\beta$ case that will be discussed in Sec. IV is found for instance in the introduction of [2]. In momentum space, the partition function may be written

The propagators of $\phi$ and $\psi$ may be read directly from the quadratic action $S_{0}$ to be $(p^{2}+m^{2})^{-1}$ and $(-ip+\mu)^{-1}$ respectively. Since the harmonic oscillator amplitude $|\langle 1|q|0\rangle|^{2}$ may be easily calculated as $(2m)^{-1}$ using either wave function or operator methods it is seen that these propagators are compatible with the exact expressions (8) and (10) in the $\lambda=0$ limit.

In the next section we will show explicitly via path integral methods that indeed the bosonic propagator is uncorrected by the Yukawa interaction $S_{I}$, and the fermionic mass $\mu$ is corrected to $\mu_{\lambda}$ in agreement with the quantum mechanical picture. The situation for a finite temperature $\beta$ will be a bit more problematic and will be the focus of Sec. IV.

The situation above involving the vanishing of all corrections to the bosonic correlation function may seem a little too glib. After all, as the order in the $\lambda$ expansion increases there are more and more potential corrections to the bosonic correlation function which appear and it seems that they must play some role in the theory. To understand the role of these diagrams we need to turn on a finite temperature so that expectation values are taken over a trace of all states and the bosonic fields can ‘see’ the fermionic sector. From the path integral perspective, the finite temperature simply modifies the $\phi$ and $\psi$ fields to be periodic and anti-periodic over the Euclidean time $\beta$, but otherwise the exact same diagrams appear as for the zero-temperature case.

Now let us briefly recap what has been shown here. We have just finished a calculation in Sec. IV.3 showing how fermion loops of arbitrary order are consistent with interpretation of $\bar{\psi}\psi$ as a number operator of a fermionic two-level system. This calculation was not specific to our model with a Yukawa interaction term. The same fermion loops appear in calculating correlation functions of number operators even in a system of free fermions, so this calculation is really a statement about how fermions work in one spacetime dimension in general.

We began considering this calculation because of the puzzle raised in Sec. IV.2. It was shown from the operator perspective in (20) that the Yukawa interaction should only correct the bosonic correlation function by a constant shift, even in the case of non-zero temperature. But before completing the calculation it was difficult to see how all the corrections to the bosonic correlation function involving fermion loops could possibly be independent of the $\tau$ parameter in the correlation function.

This puzzle involving fermion loop corrections to the bosonic correlation function was the main impetus for the considerations in this article. But as a pedagogical example which may supplement a course in quantum field theory, this need not be taken so far. Already in the introduction following Eq. 1, it was shown how canonical quantization of the natural extension of the Yukawa interaction action to $d=1$ leads to a trivially solvable quantum mechanical system. In particular the two main effects of the interaction were shifting the energy gap of the two-level system from the bare value $\mu$ to the corrected value $\mu_{\lambda}$, and shifting the ground state expectation value of the harmonic oscillator position operator to some non-zero value when the two-level system is in its $N=1$ sector.

Both of these effects were shown from the path integral perspective in Sec.III by calculating the two diagrams in Fig 1. To find the shift in the ground state expectation value the tadpole diagram was calculated in Sec.III.1. To calculate the correction to the ‘mass’ $\mu_{\lambda}$, the fermionic two-point function was calculated in Sec. III.2. It was hoped that presenting these calculations where the results have a clear quantum mechanical interpretation may improve understanding of similar calculations of vacuum expectation values and self-energy corrections in standard quantum field theory, without all the extra complications of renormalization.