External Potentials and Ehrenfest Relations in
Lagrangian Field Theories

It is well understood in classical mechanics that the conservation of linear momentum in a system only holds when no external force is acting on it. Put in the language of Noether’s theorem, a system is not symmetric under translation when a spatially dependent potential acts on it—in which case the conservation of linear momentum is broken. However, much of the richness in classical mechanics comes from understanding how mechanical momentum changes in time when such a potential is applied. This is captured by Newton’s second law,

$$\frac{d}{dt}\mathbf{p}=-\nabla V.$$

(1)

Of course, this expression reduces to a statement of momentum conservation when the potential $V$ is not spatially dependent. While this result can be easily derived as the equation of motion in a classical Lagrangian of the form $L(\mathbf{q},\dot{\mathbf{q}},t)$, there has so far been no systematic way to produce analogous results for the field momentum of a field Lagrangian $\mathcal{L}(\varphi_{1},\dots,\varphi_{n},\partial_{\mu}\varphi_{1},\dots,\partial_{\mu}\varphi_{n})$. This paper develops such a method by first considering a non-relativistic field theory that describes the Schrödinger equation with an external, coordinate-dependent potential in Section III. Conservation of energy and momentum does not hold in general for such a system, but a close examination of the potential’s response to coordinate transformations yields Ehrenfest’s theorem for linear and angular momentum as well as a similar result for energy. We then use a generalized version of the same method in Section IV to derive analogous results in a complex Klein-Gordon field theory with minimal electric potential coupling. In this paper, such results are referred to as “Ehrenfest” or “Ehrenfest-like” relations owing to their structural similarities with the original results obtained from the Schrödinger field theory. However, the content of relativistic Ehrenfest relations has an entirely different interpretation from the non-relativistic case.

In the context of classical field theory, Noether’s theorem describes a fascinating equivalence between continuous symmetries of a Lagrangian system and its conserved quantities. Starting with an action of the form

$$\mathcal{S}=\int\mathcal{L}(\varphi,\partial_{\mu}\varphi)\,d^{4}x,$$

(3)

where $\mathcal{L}$ is a Lagrangian density (henceforth: “Lagrangian”) that depends on a field $\varphi$ and its derivatives. Using the principle of stationary action, the field $\varphi$ must satisfy the Euler-Lagrange field equation of motion

$$-\partial_{\mu}\left(\frac{\partial{\mathcal{L}}}{\partial{\varphi_{,\mu}}}\right)+\frac{\partial{\mathcal{L}}}{\partial{\varphi}}=0.$$

(4)

A “symmetry” of the system is a transformation that yields the same physical predictions. A transformation of the field,

$$\varphi\to\varphi+\delta\varphi,$$

(5)

is a symmetry if it leaves Eq. (4) invariant. This holds most generally when the transformation deforms the Lagrangian by a divergence of some function $K^{\mu}$:

$$\mathcal{L}\Big{(}\varphi+\delta\varphi,\partial_{\mu}(\varphi+\delta\varphi)\Big{)}-\mathcal{L}(\varphi,\partial_{\mu}\varphi)=\partial_{\mu}K^{\mu}.$$

(6)

(We will later argue that the reverse is not true: invariance of Eq. (4) does not imply symmetry under a given transformation.) Calculating the left-hand side of Eq. (6) with a series expansion and simplifying with the equation of motion yields

$$\mathcal{L}\Big{(}\varphi+\delta\varphi,\partial_{\mu}(\varphi+\delta\varphi)\Big{)}-\mathcal{L}(\varphi,\partial_{\mu}\varphi)=\partial_{\mu}\left(\frac{\partial{\mathcal{L}}}{\partial{\varphi_{,\mu}}}\delta\varphi\right).$$

(7)

Equating this to the surface term $\partial_{\mu}K^{\mu}$ yields the vanishing divergence

$$\partial_{\mu}\left(\frac{\partial{\mathcal{L}}}{\partial{\varphi_{,\mu}}}\delta\varphi-K^{\mu}\right)=0,$$

(8)

which defines a conservation law in the form of a continuity equation. The term in parentheses defines a Noether current. Importantly, if the action depends on more than one field, we would need to add the response of each field to the transformation when defining the Noether current. In this paper, we will examine Lagrangian systems that depend on two fields which are complex conjugates of each other, so Noether currents will take the form

$$J^{\mu}\equiv\frac{\partial{\mathcal{L}}}{\partial{\varphi_{,\mu}}}\delta\varphi+\frac{\partial{\mathcal{L}}}{\partial{\overline{\varphi}_{,\mu}}}\delta\overline{\varphi}-K^{\mu}.$$

(9)

We now proceed to discuss how the conservation of these Noether currents is affected by an external scalar potential.

Suppose we are given a physical system that is influenced by an external, coordinate-dependent potential. One such example is a system described by the Schrödinger equation

$$-\frac{\hbar^{2}}{2m}\nabla^{2}\Psi+V\Psi=i\hbar\frac{\partial{\Psi}}{\partial{t}}.$$

(10)

Crucially, Eq. (10) does not specify the source or dynamics of the potential $V$, which is why we say that it is external to the theory. To formulate this equation in terms of field theory, consider a Schrödinger LagrangianFranklin2017 TakahashiRop1986

$$\mathcal{L}=\frac{\hbar^{2}}{2m}\partial_{j}\overline{\Psi}\partial^{j}\Psi+V\overline{\Psi}\Psi-\frac{i\hbar}{2}\left(\overline{\Psi}\dot{\Psi}-\Psi\dot{\overline{\Psi}}\right),$$

(11)

where the fields are $\Psi$ and $\overline{\Psi}$. The equation of motion associated with the latter,

$$-\frac{\partial{}}{\partial{t}}\left(\frac{\partial{\mathcal{L}}}{\partial{\dot{\overline{\Psi}}}}\right)-\frac{\partial{}}{\partial{x^{j}}}\left(\frac{\partial{\mathcal{L}}}{\partial{\overline{\Psi}_{,j}}}\right)+\frac{\partial{\mathcal{L}}}{\partial{\overline{\Psi}}}=0,$$

(12)

reduces to Eq. (10). To account for the potential’s response to a coordinate transformation, we treat the Lagrangian as an explicit function of $V$ in addition to the fields and their derivatives:

$$\mathcal{L}=\mathcal{L}(\Psi,\overline{\Psi},\partial_{\mu}\Psi,\partial_{\mu}\overline{\Psi},V).$$

(13)

Despite the awkward dependence on the potential, this Lagrangian still admits a $U(1)$ symmetry. Consider the complex rotation $\Psi\to e^{i\theta}\Psi$ (and cc). For small $\theta$ we have $e^{i\theta}\approx 1+i\theta$, and so the fields respond with

$$\Psi\to\Psi+i\theta\Psi\quad\text{and}\quad\overline{\Psi}\to\overline{\Psi}-i\theta\overline{\Psi}.$$

(14)

The associated conservation law becomes a statement of probability conservation,

$$\frac{\partial{}}{\partial{t}}|\Psi|^{2}+\nabla\cdot\mathbf{j}=0,$$

(15)

where $|\Psi|^{2}\equiv\overline{\Psi}\Psi$ is interpreted as a probability density and $\mathbf{j}$ is the probability current defined byGriffiths2018-vs

$$\mathbf{j}\equiv\frac{\hbar}{2mi}\left(\overline{\Psi}\nabla\Psi-\Psi\nabla\overline{\Psi}\right).$$

(16)

This result is unaffected by the presence of $V$ in the Lagrangian, but the same is not true for coordinate transformations. Consider a spacetime translation

$$x^{\mu}\to x^{\mu}+a^{\mu},$$

(17)

for some (small) constant $a^{\mu}$. The fields are scalars, so they transform as

$$\Psi\to\Psi+a^{\mu}\partial_{\mu}\Psi\quad\text{and}\quad\overline{\Psi}\to\overline{\Psi}+a^{\mu}\partial_{\mu}\overline{\Psi},$$

(18)

and crucially, so does the potential:

$$V\to V+a^{\mu}\partial_{\mu}V.$$

(19)

As a scalar, the Lagrangian also transforms as $\mathcal{L}\to\mathcal{L}+a^{\mu}\partial_{\mu}\mathcal{L}$. Thus the change in the Lagrangian (which we will denote as $\delta\mathcal{L}$) calculated by treating $\mathcal{L}$ as a scalar is given by

$$\delta\mathcal{L}=\partial_{\mu}(a^{\mu}\mathcal{L}),$$

(20)

which gives the surface term. Care must be taken to understand what is meant here: as a scalar, the Lagrangian does changes by an overall divergence, and so the equations of motion are invariant even in the presence of $V$. However, computing $\delta\mathcal{L}$ from a first-order series expansion in all arguments of the transformed Lagrangian does not yield an overall divergence, and dependence on $V$ is the culprit:

$$\delta\mathcal{L}=a^{\nu}\left[\partial_{\mu}\left(\frac{\partial{\mathcal{L}}}{\partial{\Psi_{,\mu}}}\Psi_{,\nu}+\frac{\partial{\mathcal{L}}}{\partial{\overline{\Psi}_{,\mu}}}\overline{\Psi}_{,\nu}\right)+\frac{\partial{\mathcal{L}}}{\partial{V}}\partial_{\nu}V\right].$$

(21)

We can thus see that $\mathcal{L}$ is not translationally symmetric, and hence there is no conservation law to be obtained from a vanishing divergence. That is, invariance of a potential-dependent Lagrangian under a coordinate transformation does not imply a symmetry under that transformation. However, we can still set Eqs. (20) and (21) equal to each other and see what is obtained instead of a conservation law. After discarding the overall constant $a^{\nu}$, we define the energy-momentum tensor in a standard way:

$$T^{\mu}_{\,\,\,\nu}\equiv\frac{\partial{\mathcal{L}}}{\partial{\Psi_{,\mu}}}\Psi_{,\nu}+\frac{\partial{\mathcal{L}}}{\partial{\overline{\Psi}_{,\mu}}}\overline{\Psi}_{,\nu}-\delta^{\mu}_{\,\,\,\nu}\mathcal{L}.$$

(22)

We still say that $T^{\mu}_{\,\,\,\nu}$ represents four Noether currents containing the energy ($\nu=0$) and momentum ($\nu=1,2,3$) densities stored in the fields. However, the divergence of $T^{\mu}_{\,\,\,\nu}$ is not in general zero. Instead, the derivatives of the potential act as a “source” in the continuity equations

$$\partial_{\mu}T^{\mu}_{\,\,\,\nu}=-\frac{\partial{\mathcal{L}}}{\partial{V}}\partial_{\nu}V.$$

(23)

The conservation of these currents depends on the presence of an external potential and its derivatives. In analogy with classical mechanics, the momentum of a system is conserved only when there is no coordinate-dependent external potential acting on it. When an external potential is applied, momentum is no longer conserved and the change in momentum is summarized by Newton’s second law: $d\mathbf{p}/dt=-\nabla V$. The equivalent result in non-relativistic quantum mechanics would be Ehrenfest’s theorem for momentum, $d\langle\hat{\mathbf{p}}\rangle/dt=\langle-\nabla V\rangle$. Section III.1 will show that this is one of two statements contained in Eq. (23).

The general outline of the previous section appears to be a way to generate Ehrenfest-like relations directly from a Lagrangian, without reference to the probabilistic interpretation of the wavefunction. We use this as motivation to develop analogous results in a relativistic setting using complex Klein-Gordon fields coupled with an electric potential. We use a minimal coupling given by the covariant derivative

$$D_{\mu}\equiv\partial_{\mu}-i\frac{q}{\hbar}A_{\mu},$$

(49)

where $q$ determines the coupling strength and

$$A^{\mu}\equiv\begin{pmatrix}V/c\\ \mathbf{A}\end{pmatrix}$$

(50)

is the electromagnetic four-potential (note that $V$ now has a dimension of energy per unit charge). Since we are only interested in the scalar potential term, we set $\mathbf{A}=0$. In order to treat the potential as external to the theory, we will further omit the $\frac{1}{4\mu_{0}}F^{\alpha\beta}F_{\alpha\beta}$ term used in a scalar QED action. With these modifications, the Lagrangian we use is

	$\displaystyle\mathcal{L}$	$\displaystyle=\overline{\left(D_{\mu}\phi\right)}\left(D^{\mu}\phi\right)+\frac{m^{2}c^{2}}{\hbar^{2}}\overline{\phi}\phi$
		$\displaystyle=\overline{\phi}_{,\mu}\phi_{,}^{\,\,\mu}+i\frac{qV}{\hbar c^{2}}\left(\overline{\phi}\dot{\phi}-\phi\dot{\overline{\phi}}\right)+\left(\frac{m^{2}c^{2}}{\hbar^{2}}-\frac{q^{2}V^{2}}{\hbar^{2}c^{2}}\right)\,\overline{\phi}\phi.$		(51)

Using the d’Alembertian operator $\square\equiv\partial_{\mu}\partial^{\mu}$, the field equations are

	$\displaystyle-\square\overline{\phi}+\left(\frac{m^{2}c^{2}}{\hbar^{2}}-\frac{q^{2}V^{2}}{\hbar^{2}c^{2}}\right)\overline{\phi}+2i\frac{qV}{\hbar c^{2}}\dot{\overline{\phi}}+\frac{iq\dot{V}}{\hbar c^{2}}\overline{\phi}=0$		(52)
	$\displaystyle-\square\phi+\left(\frac{m^{2}c^{2}}{\hbar^{2}}-\frac{q^{2}V^{2}}{\hbar^{2}c^{2}}\right)\phi-2i\frac{qV}{\hbar c^{2}}\dot{\phi}-\frac{iq\dot{V}}{\hbar c^{2}}\phi=0.$		(53)

As before, we will first find the conservation law associated with a $U(1)$ symmetry, $\phi\to e^{i\theta}\phi$. The conserved current is

$$J^{\mu}=i\theta\left(\overline{\phi}\partial^{\mu}\phi-\phi\partial^{\mu}\overline{\phi}-2\frac{iq}{\hbar}A^{\mu}\overline{\phi}\phi\right).$$

(54)

The 0-component can be simplified by separating the fields into their spatial and time-dependent components

$$\phi=\psi(\mathbf{r})e^{-iEt/\hbar},$$

(55)

and the complex conjugate. This allows us to write the time derivatives of the fields as

$$\dot{\phi}=\frac{E}{i\hbar}\phi\quad\text{and}\quad\dot{\overline{\phi}}=-\frac{E}{i\hbar}\overline{\phi}.$$

(56)

After setting units, the conservation law reads

$$\frac{\partial{}}{\partial{t}}\rho+\nabla\cdot\mathbf{j}=0,$$

(57)

where

$$\rho\equiv\frac{2q}{\hbar^{2}c^{2}}(E-qV)\overline{\phi}\phi$$

(58)

is interpreted as an electric charge density,Winter1959 and

$$\mathbf{j}\equiv i\frac{q}{\hbar}(\phi\nabla\overline{\phi}-\overline{\phi}\nabla\phi)$$

(59)

is the corresponding electric current. Since electric charge conservation appears precisely where probability conservation appeared in the Schrödinger field theory, we anticipate the same replacement to appear in the continuity equations associated with the energy-momentum tensor. Section IV.1 shows that this is indeed the case.

This paper examined the effect of external scalar potentials on Lagrangian systems. When the potential is coordinate-dependent, such systems are not symmetric under a coordinate transformation and hence conservation of energy and momentum do not hold. By accounting for the response of the potential to spacetime translation, however, we find a continuity equation wherein the derivatives of the potential act as a source of field energy and field momentum. In integral form, these continuity equations express familiar Ehrenfest relations directly from a Schrödinger Lagrangian. This approach makes no appeal to the probabilistic interpretation of the wavefunction, nor does it require explicit reference to the Schrödinger equation. We thus postulate that Eq. (23) is a general way to derive Ehrenfest-like relations for a Lagrangian acted on by an external, coordinate-dependent scalar potential.

We then demonstrated one example of Ehrenfest-like relations for a complex Klein-Gordon Lagrangian coupled to an electric potential. These results for relativistic spin 0 particles have an electric charge density $\rho$ taking the place of the probability density $|\Psi|^{2}$ from the Schrödinger Lagrangian. A natural progression of this work would be to replicate these results for a Dirac Lagrangian describing spin 1/2 particles.

It should be noted that the derivation leading to Eq. (23) is not the only way to derive the Ehrenfest relations shown this paper, although other methods are far less elegant. One could instead explicitly calculate $\partial_{\mu}T^{\mu}_{\,\,\,\nu}$ using Eqs. (24–27) for the Schrödinger tensor and Eqs. (64–67) for the Klein-Gordon tensor. This process is much more involved and requires extensive use of the field equations in order to arrive at the desired continuity equations. However, the correspondence of both methods lends support to the notion of treating a Lagrangian as an explicit function of an external scalar potential when one is applied.

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