# About non-relativistic quantum mechanics and electromagnetism.

Ladislaus Alexander Bányai* and Mircea Bundaru

###### Abstract

We describe here the coherent formulation of electromagnetism in the non-
relativistic quantum-mechanical many-body theory of interacting charged
particles. We use the mathematical frame of the field theory and its
quantization in the spirit of the QED. This is necessary because a manifold of
misinterpretations emerged especially regarding the magnetic field and gauge
invariance. The situation was determined by the historical development of
quantum mechanics, starting from the Schrödinger equation of a single particle
in the presence of given electromagnetic fields, followed by the many-body
theories of many charged identical particles having just Coulomb interactions.
Our approach to the non-relativistic QED emphasizes the role of the gauge-
invariance and of the external fields. We develop further the $1/c^{2}$
approximation of this theory allowing a closed description of the interacting
charged particles without photons. The resulting Hamiltonian coincides with
the quantized version of the Darwin Hamiltonian containing besides the Coulomb
also a current-current diamagnetic interaction. We show on some examples the
importance of this extension of the many-body theory.

Institut für Theoretische Physik, Goethe-Universität, Frankfurt am Main, Max-
von-Laue-Strasse 1, D-60438, Germany*

<EMAIL_ADDRESS>corresponding author

National Institute of Physics and Nuclear Engineering - Horia Hulubei, PO-Box
MG-6, Bucharest-Magurele, Romania**

<EMAIL_ADDRESS>
Keywords: non-relativistic QED; many-body theories; field theory;
quantization; gauge invariance; Coulomb gauge; external fields; $1/c^{2}$
approximation; current-current interaction; ideal diamgnetism; linear
response; thermal noise

##  1 Introduction

The quantum mechanics of a single charged particle in the presence of applied
electric and magnetic fields is based on the classical mechanics of a point-
like charge derived from a Lagrange function, followed by a quantization
according to the ”Poisson-bracket into commutator” recipe. This scheme however
failed for the electromagnetic interaction between charged particles. The
classical electromagnetic theory of point-like charges has neither a
Lagrangian nor Hamiltonian formulation. One eliminates by hand from the
Lorentz force the action of the fields created by the particle itself.
Nevertheless, one took over this recipe in order to include at least the
Coulomb interaction between the particles. The constructed Hamiltonian lead to
big successes as well in the theory of atomic structure, as in the solid-state
theory. Of course, one had to include also the symmetry or anti-symmetry of
the wave functions due to the bosonic, or fermionic nature of identical
particles, as well as the spin, whose origin stems actually from the
relativistic theory. This state of the solid-state theory however was unable
to include the magnetic field and forces created by the charged particles
themselves. The importance of this problem was waved out by an argument
relying on the small velocities of the electrons and ions in matter as to be
compared to the light velocity $c$. This argumentation is however false, since
it is well-known that a slow flow of a macroscopic number of electrons may
create enormous magnetic fields.

A first attempt to construct a Lagrangian and Hamiltonian including velocity-
dependent forces up to $1/c^{2}$ in the classical theory of interacting
charged point-like particles was made by Darwin Ref.[1] 100 years ago. His
idea was to neglect the retardation from the Lenart-Wichert potentials in
order to eliminate them in favor of the particle velocities and ignoring the
terms containing the velocity of the same particle. Later Landau and Lifshitz
Ref. [2] as well as Jackson [3] realized, that one may not pursue this way
without fixing the gauge. Both used implicitly or explicitly the Coulomb gauge
in their derivation. This approach based mostly on physical intuition still
does not offer a sound basis for a theory of interacting charged particles,
since the so obtained Hamiltonian again cannot be integrated in a correct
Lagrangian formulation due to the nonlinear velocity dependent forces it
implies, as well as to the artificial exclusion of the self-interaction. It is
therefore not surprising that the Darwin model had no impact on the quantum-
mechanical many-body theories of condensed matter developed in the second half
of the 20th century.

This is a very strange situation, since already in the mid 20th century
Quantum Electrodynamics (QED) was successfully developed and its is considered
as the basic theory for the relativistic, quantum mechanical description of
electrons and positrons interacting with photons. Therefore, at least a non-
relativistic quantum-mechanical description of electromagnetic interacting
electrons is desirable as an approximation to the QED. Such a reduction
however seems illusory and again leads into a deadlock, since without positive
charges no matter is stable. The condensed matter we have to deal is made up
of electrons and nuclei (not positrons!). The nuclei are not elementary
particles in the present understanding of the quantum field theory, but
composite objects made out of protons an neutrons bound together by non-
electromagnetic forces. Even worse, the protons and neutrons themselves are
not the ultimate basic pieces. A certain skepticism is justified regarding a
derivation of condensed matter theory from the standard model of interacting
elementary particles.

After renouncing to the ill-defined classical theory of point-like charges one
has to build a consistent electromagnetic theory of arbitrary charged quantum
mechanical particles by following the mathematical pattern of the QED. Such a
theory is compulsory non-relativistic, since it does not include
antiparticles. Quantization of relativistic wave equations however implies
antiparticles!

The non-relativistic QED (including the relativistic concept of the spin and
its magnetic moment) was not only formulated, but also widely used in quantum
optics, where one treats photons interacting with atoms, molecules or solids.
(See for example a recent presentation in Ref. [4].)

In the same time, most theories of condensed matter still use many-body
formulations including only Coulomb forces between the charged particles,
while one avoids the original configuration-space formulation that excludes
self-interaction. Nevertheless, by the second quantization on introduces self-
interaction implicitly again.

This rather unsatisfactory situation is due partly to the wide split between
the solid state theorists and elementary particle theorists. The purpose of
this paper (partly discussed in the book [5]) is to provide a clear Lagrangian
presentation of the non-relativistic QED of charged particles and within this
frame the derivation of a consistent reduced Hamiltonian valid up to order
$1/c^{2}$ containing only the electromagnetically interacting charged
particles and no photons. This derivation [6] is based essentially on the old
paper of Holstein, Norton and Pincus [7] about the transverse photon exchange
being at the origin of diamagnetism. It occurs that this Hamiltonian is
nothing else as the quantum mechanical version of the Darwin Hamiltonian [1]
in Jackson’s formulation [3].

We start with the classical field theory of electromagnetic fields interacting
with a Schrödinger electron obeying the Maxwell equations. This approach is
analogous to the starting point of the relativistic QED, which is based on the
Dirac equation. The next step is the construction of the Lagrangian, which is
not uniquely defined.

An essential role in any field theory play the external sources. Without these
macroscopic, classical entities, no measurement may be considered.
Unfortunately, a certain confusion is still spread in the literature about the
internal and external electromagnetic fields.( See however the 50 years old
book of Zubarev [8], where he already insisted on the importance of the
general distinction between external and total fields in the frame of the
linear response.) We insist here on this very important aspect since it is
essential for the interpretation of experiments.

There are two different ways to introduce the external fields or sources. The
first considers the ”internal” e.m. fields produced by the electron that
interacts also with ”external” fields (produced by external sources) by the
minimal e.m. coupling and it gives rise to an explicitly gauge invariant
Lagrange density. This version is used to build up a Hamiltonian. Since the
Lagrangian is ”singular” one needs to fix the gauge, namely by choosing the
Coulomb one. This choice of gauge eliminates the spurious degrees of freedom,
allowing to avoid the problems raised by Dirac’s theory of canonical formalism
[9, 10] in the presence of constraints. The quantization of this Hamiltonian
leads to the usual non-relativistic QED. The gauge freedom here is restricted
to the transformations of the external fields. This aspect is ignored in all
the discussions of magnetic effects in solid state.

The second way is to introduce explicitly a coupling of the e.m. fields
directly to the external sources. The resulting electromagnetic fields are the
”total” ones. The external electromagnetic potentials are eliminated in favor
of their sources. The Lagrange density is not any more gauge invariant, but
the action is still gauge invariant. This way is used in the functional
integral formulation of the QED, which is defined just by the Lagrange
density. No Hamiltonian is needed at all. The two approaches are equivalent,
at least for simple connected systems.

Thereafter we discuss the quantization of the first variant i.e. the non-
relativistic QED in the Hamiltonian formulation and include also the
ingredient of the spin-magnetic moment taken over from the relativistic QED.

We describe also the $1/c^{2}$ approximation of the non-relativistic QED in
the absence of photons suitable for condensed matter theory. Within this frame
already the magnetic interaction between the charged particles is taken into
account.

While formally we considered here just ”electrons” the theory may be extended
trivially to a system of negatively and positively charged particles
(electrons with ions or electrons in conduction band with holes in the valence
band).

It is worth to mention here, that unlike in the relativistic QED, the
divergences in the adiabatic perturbation theory of the S matrix were not yet
treated systematically. The many-body theory of solid-state is used always in
its cut-off form and not by a renormalization of the bare parameters. Helpful
here are the ”natural” cut-off parameters like the Debye wave-length or the
band width in a crystal.

Further, we describe the important role of this improved many-body theory for
electromagnetic linear response, superconductivity and thermal noise. The
transverse dielectric function (or conductivity) is meaningless in a pure
Coulomb theory. Only the magnetic field created by the electrons may explain
perfect diamagnetism shown in the Meissner effect. The thermal noise of
electromagnetic origin must distinguish between the longitudinal /transverse
fields or photon number measurements. All these fields need to consider
magnetic forces between the charged particles, at least in the $1/c^{2}$
approximation.

##  2 Classical field theory.

In the classical field-theory one defines the action ${\cal A}$

${\cal A}=\int d{\vec{x}}\int dt{\cal L}({\vec{x}},t)$ (1)

by a Lagrange density ${\cal L}({\vec{x}},t)$ depending on some fields
$\phi_{i}({\vec{x}},t)$ together with their first time and space derivatives.
The variational principle $\delta A=0$ gives rise to the generalized Euler-
Lagrange equations

$\frac{\partial}{\partial t}\frac{\delta{\cal
L}}{\delta{\dot{\phi}_{i}({\vec{x}},t)}}+\frac{\partial}{\partial
x_{\mu}}\frac{\delta{\cal
L}}{\delta\frac{\partial\phi_{i}({\vec{x}},t)}{\partial
x_{\mu}}}-\frac{\delta{\cal L}}{\delta\phi_{i}({\vec{x}},t)}=0\enspace.$ (2)

Here the symbol $\partial$ means ordinary derivative, while the symbol
$\delta$ means functional derivative. Two Lagrangian densities that differ by
the time derivative or by the divergence of a function give rise to the same
action and therefore are considered to be equivalent, taking into account the
vanishing of the fields at infinity.

The generalized canonical conjugate momenta for the fields
$\phi_{i}({\vec{x}},t)$ are defined by

$\Pi_{\phi_{i}}=\frac{\delta{\cal L}}{\delta{\dot{\phi}_{i}}}$ (3)

and the Hamiltonian density is

${\cal H}(\phi,\Pi_{\phi})=-{\cal L}+\Pi_{\phi_{i}}{\dot{\phi}_{i}}\enspace,$
(4)

provided no relations (constraints) appear between the canonical conjugate
momenta. Lagrangians with constraints however have to be handled with Dirac’s
canonical formalism [9, 10], that implies also a redefinition of the Poisson
bracket.

##  3 Classical Maxwell fields coupled to a quantum mechanical electron and
external sources.

The classical Maxwell equations are two with sources $\rho$ and $\vec{j}$

$\displaystyle\nabla\times\vec{B}$ $\displaystyle=$
$\displaystyle\frac{4\pi}{c}\vec{j}+\frac{1}{c}\frac{\partial}{\partial
t}\vec{E}$ (5) $\displaystyle\nabla\vec{E}$ $\displaystyle=$ $\displaystyle
4\pi\rho$ (6)

and two without sources

$\displaystyle\nabla\vec{B}$ $\displaystyle=$ $\displaystyle 0$ (7)
$\displaystyle\nabla\times\vec{E}$ $\displaystyle=$
$\displaystyle-\frac{1}{c}\frac{\partial}{\partial t}\vec{B}\enspace$ (8)

(we use Gaussian units like Ref. [2] ). The equations without sources are
automatically satisfied by the introduction of the electromagnetic potentials

$\displaystyle\vec{B}$ $\displaystyle=$ $\displaystyle\nabla\times\vec{A}$ (9)
$\displaystyle\vec{E}$ $\displaystyle=$ $\displaystyle-\nabla
V-\frac{1}{c}\frac{\partial}{\partial t}\vec{A}\enspace.$ (10)

Therefore we may concentrate on Eqs. 5 and 6 depending on the sources.

We are looking for the ”internal” electromagnetic fields produced by a single
quantum mechanical electron. Thus the sources $\rho$ and ${\vec{j}}$ are given
by the quantum mechanical charge and current of an electron described by the
wave function $\psi({\vec{x}},t)$ (for simplicity without spin) coupled to the
(”internal”) electromagnetic fields $\vec{A}$, $V$ as well as to some
”external” classical fields $\vec{A}_{ext}$, $V_{ext}$. The later are supposed
to satisfy their own Maxwell equations with the external macroscopic sources
$\rho_{ext}$ and $\vec{j}_{ext}$.

Thus

$\displaystyle\rho({\vec{x}},t)$ $\displaystyle=$ $\displaystyle
e\psi({\vec{x}},t)^{*}\psi({\vec{x}},t)$ (11)
$\displaystyle\vec{j}({\vec{x}},t)$ $\displaystyle=$
$\displaystyle\frac{e}{2m}\psi({\vec{x}},t)^{*}\left(-\imath\hbar\nabla+\frac{e}{c}({\vec{A}}({\vec{x}},t)+{\vec{A}}_{ext}({\vec{x}},t))\right)\psi({\vec{x}},t)+c.c$
(12)

while the wave function $\psi({\vec{x}},t)$ satisfies the Schrödinger equation

$\imath\hbar\frac{\partial}{\partial
t}\psi({\vec{x}},t)=\left(\frac{1}{2m}\left(-\imath\hbar\nabla+\frac{e}{c}(\vec{A}(x,t)+\vec{A}_{ext})\right)^{2}+e\left(V({\vec{x}},t)+V_{ext}(\vec{x},t)\right)\right)\psi({\vec{x}},t)\enspace.$
(13)

Using only this equation one gets the continuity equation

$\nabla\vec{j}+\frac{\partial}{\partial t}\rho=0$ (14)

required for the consistency of Eqs. 5 and 6.

The Schrödinger equation that couples to the e.m. fields, as well as the
current density were introduced here according to the minimal recipe

$\displaystyle\frac{\hbar}{\imath}\nabla\psi$ $\displaystyle\to$
$\displaystyle\left(\frac{\hbar}{\imath}\nabla+\frac{e}{c}({\vec{A}}+{\vec{A}}_{ext})\right)\psi$
(15) $\displaystyle\frac{\hbar}{\imath}\frac{\partial}{\partial t}\psi$
$\displaystyle\to$
$\displaystyle\left(\frac{\hbar}{\imath}\frac{\partial}{\partial
t}+e(V+eV_{ext})\right))\psi\enspace.$ (16)

This is a peculiar case of the covariant (here abelian) Yang-Mills derivative.

An alternative formulation is to define the ”total” electromagnetic potentials
and fields as those produced by the electron and the external sources

$\vec{A}^{\prime}\equiv\vec{A}+\vec{A}_{ext};\qquad V^{\prime}\equiv
V+V_{ext}\enspace,$ (17)

which are usually denoted with the same symbol as the internal ones that may
lead to confusions. Their sources are given by the sum of the electron charge
and current densities and the external charge and current densities. Then no
external fields, but just their sources appear in the theory. However it is
very important to discern the two descriptions, although formally they are
equivalent.

The electromagnetic potentials and the wave function are not uniquely defined,
they allow a simultaneous gauge transformation

$\displaystyle V({\vec{x}},t)$ $\displaystyle\to$ $\displaystyle
V({\vec{x}},t)+\frac{1}{c}{\dot{\chi}({\vec{x}},t)}$ (18)
$\displaystyle\vec{A}({\vec{x}},t)$ $\displaystyle\to$
$\displaystyle\vec{A}({\vec{x}},t)-\nabla\chi({\vec{x}},t)$
$\displaystyle\psi({\vec{x}},t)$ $\displaystyle\to$
$\displaystyle\psi({\vec{x}},t)e^{-\frac{\imath e}{\hbar c}\chi({\vec{x}},t)}$

with an arbitrary differentiable function $\chi({\vec{x}},t)$ that do not
change neither the Maxwell fields ${\vec{B}}$, ${\vec{E}}$, their sources
$\rho$, ${\vec{j}}$, nor the whole system of equations from Eq.5 to Eq.14. A
similar gauge invariance is true with respect to the independent gauge
transformations of the external potentials.

This arbitrariness is not unexpected, since we actually introduced 4 degrees
of freedom instead of the 3 independent degrees of freedom (the transverse
magnetic field and the longitudinal electric field).

##  4 Classical Lagrange density

The first problem is then to find a Lagrangian giving rise to these equation
in terms of the fields $V({\vec{x}},t)$, ${\vec{A}}({\vec{x}},t)$ and
$\psi({\vec{x}},t)$ as dynamical variables (generalized coordinates).

It is easy to see, that the uncoupled Maxwell

$\displaystyle\nabla\times\vec{B}$ $\displaystyle=$
$\displaystyle\frac{1}{c}\frac{\partial}{\partial t}\vec{E}$ (19)
$\displaystyle\nabla\vec{E}$ $\displaystyle=$ $\displaystyle 0$ (20)

and Schrödinger equations

$\imath\hbar\frac{\partial}{\partial
t}\psi({\vec{x}},t)=-\frac{\hbar^{2}}{2m}\nabla^{2}\psi({\vec{x}},t)$ (21)

follow from the free Lagrangian density

$\displaystyle{\cal L}_{0}({\vec{x}},t)$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi}\left(\nabla
V({\vec{x}},t)+\frac{1}{c}{\dot{\vec{A}}}({\vec{x}},t)\right)^{2}-\frac{1}{8\pi}\left(\nabla\times{\vec{A}}({\vec{x}},t)\right)^{2}$
$\displaystyle-$
$\displaystyle\frac{\hbar^{2}}{2m}\nabla\psi^{*}({\vec{x}},t)\nabla\psi({\vec{x}},t)-\frac{\imath\hbar}{2}\left({\dot{\psi}}({\vec{x}},t)^{*}\psi({\vec{x}},t)-\psi({\vec{x}},t)^{*}{\dot{\psi}}({\vec{x}},t)\right)\enspace.$

The introduction of the e.m. fields by the minimal recipe Eqs. 15 and 16 into
the free Lagrangian ${\cal L}_{0}$ leads to the Lagrange density

$\displaystyle{\cal L}({\vec{x}},t)$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi}\left(\nabla V+\frac{1}{c}\frac{\partial}{\partial
t}{\vec{A}}\right)^{2}-\frac{1}{8\pi}\left(\nabla\times{\vec{A}}\right)^{2}$
$\displaystyle-$
$\displaystyle\frac{1}{2m}\left(-\frac{\hbar}{\imath}\nabla+\frac{e}{c}({\vec{A}}+{\vec{A}}_{ext})\right)\psi^{*}\left(\frac{\hbar}{\imath}\nabla+\frac{e}{c}({\vec{A}}+{\vec{A}}_{ext})\right)\psi$
$\displaystyle-$
$\displaystyle\frac{1}{2}\psi^{*}\left(\frac{\hbar}{\imath}\frac{\partial}{\partial
t}+e(V+V_{ext}\right)\psi-\frac{1}{2}\psi\left(-\frac{\hbar}{\imath}\frac{\partial}{\partial
t}+e(V+V_{ext})\right)\psi^{*}\enspace,$

that on his turn gives rise to the Maxwell Eqs. 5 and 6 as well as to the
Schrödinger equation 13. This Lagrange density ${\cal L}$ is by construction
invariant against gauge transformations of the fields ${\vec{A}}$, $V$ and/or
${\vec{A}}_{ext}$, $V_{ext}$.

On the other hand, if one works with the ”total” potentials
$\vec{A}^{\prime}$, $V^{\prime}$ of Eq. 17 the Lagrange density looks as

$\displaystyle{\cal L}^{\prime}({\vec{x}},t)$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi}\left(\nabla
V^{\prime}+\frac{1}{c}\frac{\partial}{\partial
t}{\vec{A}}^{\prime}\right)^{2}-\frac{1}{8\pi}\left(\nabla\times{\vec{A}}^{\prime}\right)^{2}$
$\displaystyle-$
$\displaystyle\frac{1}{2m}\left(-\frac{\hbar}{\imath}\nabla+\frac{e}{c}{\vec{A}}^{\prime}\right)\psi^{*}\left(\frac{\hbar}{\imath}\nabla+\frac{e}{c}{\vec{A}}^{\prime}\right)\psi$
$\displaystyle-$
$\displaystyle\frac{1}{2}\psi^{*}\left(\frac{\hbar}{\imath}\frac{\partial}{\partial
t}+eV^{\prime}\right)\psi-\frac{1}{2}\psi\left(-\frac{\hbar}{\imath}\frac{\partial}{\partial
t}+eV^{\prime}\right)\psi^{*}$ $\displaystyle+$ $\displaystyle
V^{\prime}({\vec{x}},t)\rho_{ext}({\vec{x}},t)+\frac{1}{c}{\vec{A}}^{\prime}({\vec{x}},t){\vec{j}}_{ext}({\vec{x}},t)\enspace.$

This alternative Lagrange density is not explicitly gauge invariant, but the
corresponding action Eq.1 is still gauge invariant. This may be shown after
partial integration by using the continuity equation of the external sources.

This version is used in the functional (path) integral formulation of the QED
in the whole homogeneous space to define the generating functional of the
Greeen functions. This formulation of the QED uses only the Lagrangian and
needs no definition of any Hamiltonian. The two versions are however not
completely equivalent. In what follows we shall develop a Hamiltonian
formalism in the Fock space starting from the Lagrangian density Eq. 4 that is
manifestly gauge invariant. This formulation is appropriate to include
boundary conditions and multiple-connectivity.

##  5 The classical Hamiltonian in the Coulomb gauge.

In this Section we build the classical Hamiltonian out of the Lagrangian Eq.
4, needed for an operator formulation of the non-relativistic QED.
Unfortunately, this Lagrangian density is a so called singular one. The time
derivative of the variable $V$ is not present in them and therefore the
corresponding canonical momentum is vanishing i.e. we have a constraint in the
canonical formalism. Lagrangians with constraints, as we already mentioned,
have to be handled with Dirac’s canonical formalism [9, 10], that implies also
a redefinition of the Poisson bracket. The simplest way out is however to use
the choice of the gauge in such a way as to eliminate the spurious degrees of
freedom from the Lagrangian before we could construct a Hamiltonian.

The Coulomb gauge defined by

$\nabla{\vec{A}}({\vec{x}},t)=0$ (25)

leaves only the physical transverse degrees of freedom of the photons and
simultaneously eliminates the scalar potential in favor of the electron charge
density

$V({\vec{x}},t)=\int
d{\vec{x}}^{\prime}\frac{\rho({\vec{x}}^{\prime},t)}{|{\vec{x}}-{\vec{x}}^{\prime}|}\enspace.$
(26)

We shall construct the Hamiltonian the usual way, however taking into account
of the above constraint and define the canonical conjugate momenta as

$\displaystyle\Pi_{\psi}$ $\displaystyle\equiv$
$\displaystyle\frac{\delta{\cal
L}}{\delta{\dot{\psi}}}=\frac{\imath\hbar}{2}\psi^{*}$ (27)
$\displaystyle\Pi_{\psi^{*}}$ $\displaystyle\equiv$
$\displaystyle\frac{\delta{\cal
L}}{\delta{\dot{\psi}^{*}}}=-\frac{\imath\hbar}{2}\psi$ (28)
$\displaystyle\Pi^{\mu}_{{A}_{\mu}}$ $\displaystyle\equiv$
$\displaystyle\frac{{\cal L}}{\delta{\dot{A}}^{\mu}}=\frac{1}{4\pi
c}\left(\frac{\partial}{x_{\mu}}V+\frac{1}{c}{\dot{A}}^{\mu}\right);\quad(\mu=1^{,}2,3)$
(29) $\displaystyle\Pi_{V}$ $\displaystyle\equiv$ $\displaystyle 0\enspace.$
(30)

and the Hamiltonian density is

${\cal H}=-{\cal
L}+{\vec{\Pi}}_{{\vec{A}}}\dot{\vec{A}}+\Pi_{\psi}{\dot{\psi}}+\Pi_{\psi^{*}}{\dot{\psi}^{*}}\enspace.$
(31)

Using the notation $\vec{A}_{\bot}$ for the vector potential in the Coulomb
gauge we have explicitly

$\displaystyle{\cal H}$ $\displaystyle=$
$\displaystyle-\frac{1}{8\pi}\left(\nabla
V+\frac{1}{c}{\dot{\vec{A}}_{\bot}}\right)^{2}+\frac{1}{8\pi}\left(\nabla\times{\vec{A}}_{\bot}\right)^{2}+\frac{1}{4\pi
c}{\dot{\vec{A}}_{\bot}}\left(\nabla V+\frac{1}{c}{\dot{\vec{A}}\bot}\right)$
$\displaystyle+$
$\displaystyle\frac{1}{2m}\left(-\frac{\hbar}{\imath}\nabla\psi^{*}+\frac{e}{c}({\vec{A}}_{\bot}+{\vec{A}}_{ext})\psi^{*}\right)\left(\frac{\hbar}{\imath}\nabla\psi+\frac{e}{c}({\vec{A}}_{\bot}+{\vec{A}}_{ext})\psi\right)+e(V+V_{ext})\psi^{*}\psi$

or

$\displaystyle{\cal H}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi}\left(\nabla
V+\frac{1}{c}{\dot{\vec{A}}_{\bot}}\right)^{2}+\frac{1}{8\pi}\left(\nabla\times{\vec{A}}_{\bot}\right)^{2}-\frac{1}{4\pi}\nabla
V\left(\nabla V+\frac{1}{c}{\dot{\vec{A}}_{\bot}}\right)$ $\displaystyle+$
$\displaystyle\frac{1}{2m}\left(-\frac{\hbar}{\imath}\nabla\psi^{*}+\frac{e}{c}({\vec{A}}_{\bot}+{\vec{A}}_{ext})\psi^{*}\right)\left(\frac{\hbar}{\imath}\nabla\psi+\frac{e}{c}({\vec{A}}_{\bot}+{\vec{A}}_{ext})\psi\right)+e(V+V_{ext})\psi^{*}\psi\enspace.$

In the Hamiltonian

$H\equiv\int d{\vec{x}}{\cal H}({\vec{x}})$ (34)

one may use a partial integration in order to obtain

$\displaystyle H$ $\displaystyle=$ $\displaystyle\int
d{\vec{x}}\left[\frac{1}{8\pi}\left(\nabla
V+\frac{1}{c}{\dot{\vec{A}}_{\bot}}\right)^{2}+\frac{1}{8\pi}\left(\nabla\times{\vec{A}}_{\bot}\right)^{2}+\frac{1}{4\pi}V\nabla\left(\nabla
V+\frac{1}{c}{\dot{\vec{A}}}\right)\right.$ $\displaystyle+$
$\displaystyle\left.\frac{1}{2m}\left(-\frac{\hbar}{\imath}\nabla\psi^{*}+\frac{e}{c}({\vec{A}}_{\bot}+\vec{A}_{ext})\psi^{*}\right)\left(\frac{\hbar}{\imath}\nabla\psi+\frac{e}{c}({\vec{A}}_{\bot}+\vec{A}_{ext})\psi\right)+e(V+V_{ext})\psi^{*}\psi\right]\enspace.$

Due to the transversality of the vector potential and expressing the scalar
potential through the total charge density Eq. 26 one gets further

$\displaystyle H$ $\displaystyle=$ $\displaystyle\int
d{\vec{x}}\left[\frac{1}{8\pi}\left(\nabla
V+\frac{1}{c}{\dot{\vec{A}}_{\bot}}\right)^{2}+\frac{1}{8\pi}\left(\nabla\times{\vec{A}}_{\bot}\right)^{2}\right.$
$\displaystyle+$
$\displaystyle\left.\frac{1}{2m}\left(-\frac{\hbar}{\imath}\nabla\psi^{*}+\frac{e}{c}({\vec{A}}_{\bot}+\vec{A}_{ext})\psi^{*}\right)\left(\frac{\hbar}{\imath}\nabla\psi+\frac{e}{c}({\vec{A}}_{\bot}+\vec{A}_{ext})\psi\right)+eV_{ext}\psi^{*}\psi\right]\enspace,$

or

$\displaystyle H\\!$ $\displaystyle=$
$\displaystyle\\!\\!\int\\!\\!d{\vec{x}}\left[\frac{1}{8\pi}\\!\left({\vec{E}}^{2}+{\vec{B}}^{2}\right)+\frac{1}{2m}\\!\left(-\frac{\hbar}{\imath}\nabla\psi^{*}+\frac{e}{c}({\vec{A}}_{\bot}+{\vec{A}}_{ext})\psi^{*}\right)\left(\frac{\hbar}{\imath}\nabla\psi+\frac{e}{c}({\vec{A}}_{\bot}+{\vec{A}}_{ext})\psi\right)\right.$
(37) $\displaystyle+$ $\displaystyle\left.V_{ext}\psi^{*}\psi\right]\enspace.$

Apparently the Coulomb interaction $eV\psi^{*}\psi$ disappeared, but actually
it is contained properly in the energy of the longitudinal electric field.
Since under the integral

${\vec{E}}^{2}=(\nabla V)^{2}+(\frac{1}{c}{\dot{\vec{A}}_{\bot}})^{2}$

and after a partial integration

$(\nabla V)^{2}\to-V\nabla^{2}V$

we get

$\displaystyle H$ $\displaystyle=$ $\displaystyle\int
d{\vec{x}}\left[\frac{1}{8\pi}\\!\left({\vec{E}}_{\bot}^{2}+{\vec{B}}^{2}\right)+e\psi^{*}\psi(\frac{1}{2}V+V_{ext})\right.$
$\displaystyle+$
$\displaystyle\left.\frac{1}{2m}\left(-\frac{\hbar}{\imath}\nabla\psi^{*}+\frac{e}{c}({\vec{A}}_{\bot}+{\vec{A}}_{ext})\psi^{*}\right)\left(\frac{\hbar}{\imath}\nabla\psi+\frac{e}{c}({\vec{A}}_{\bot}+{\vec{A}}_{ext})\psi\right)\right]\enspace.$

The first term represents the energy of the transverse ”photon” (radiation-)
field.

Obviously, while the ”internal” potentials are already fixed, one has still a
restricted gauge invariance of the Hamiltonian against the gauge
transformations of the external potentials.

##  6 Quantization.

Starting from our classical Hamiltonian in Coulomb gauge Eq.5, after the usual
equal-time quantization of the anti-commuting electron wave functions

$[\psi({\vec{x}},t),\psi^{+}({\vec{x}}^{\prime},t)]_{+}=\delta({\vec{x}}-{\vec{x}}^{\prime})$

and the introduction of creation and annihilation operators
$b_{\vec{q},\lambda}^{+}$ and $b_{\vec{q},\lambda}$ of photons of polarization
$\lambda$ and momentum ${\vec{q}}$ one defines the quantized transverse e.m.
vector potential

$\vec{A}_{\bot}(\vec{x})=\sum_{\lambda=1,2}\sqrt{\frac{\hbar
c}{\Omega}}\sum_{\vec{q}}\frac{1}{\sqrt{|\vec{q}|}}\vec{e}_{\vec{q}}^{(\lambda)}e^{-\imath\vec{q}\vec{x}}\left(b_{\vec{q},\lambda}+b_{-\vec{q},\lambda}^{+}\right)$
(39)

taken with periodical boundary conditions. This definition brings the photon
part of the Hamiltonian to a diagonal form. Here the bosonic commutators are

$\left[b_{\vec{q},\lambda},b_{\vec{q}\;^{\prime},\lambda^{\prime}}^{+}\right]=\delta_{\vec{q},\vec{q}\;^{\prime}}\delta_{\lambda\lambda^{\prime}}$

and the unit vectors $\vec{e}_{\vec{q}}^{(\lambda)}$ are orthogonal to the
wave vector $\vec{q}$ and to each other

$\vec{q}\vec{e}_{\vec{q}}^{(\lambda)}=0\quad;\qquad\vec{e}_{\vec{q}}^{(\lambda)}\vec{e}_{\vec{q}}^{(\lambda^{\prime})}=\delta_{\lambda\lambda^{\prime}}\quad;\qquad\vec{e}_{\vec{q}}^{(\lambda)}=\vec{e}_{-\vec{q}}^{(\lambda)}\quad;\qquad(\lambda,\lambda^{\prime}=1,2)\enspace.$

With these ingredients and normal ordering the operators one gets the non-
relativistic QED Hamiltonian

$\displaystyle
H^{QED}=\sum_{\vec{q},\lambda}\hbar\omega_{q}b_{\vec{q},\lambda}^{+}b_{\vec{q},\lambda}$
(40)
$\displaystyle+\int\\!d{\vec{x}}N\\!\left[\frac{1}{2m}\\!\left(\imath\hbar\nabla\psi^{+}({\vec{x}})+\frac{e}{c}({\vec{A}}_{\bot}({\vec{x}})+{\vec{A}}_{ext}({\vec{x}},t))\psi^{+}({\vec{x}})\right)\times\right.$
$\displaystyle\left.\left(\frac{\hbar}{\imath}\nabla\psi({\vec{x}},t)+\frac{e}{c}({\vec{A}}_{\bot}({\vec{x}})+{\vec{A}}_{ext}({\vec{x}},t)))\psi({\vec{x}})\right)\right]$
$\displaystyle+\frac{1}{2}\int d\vec{x}\int
d{\vec{x}}^{\prime}\psi^{+}(\vec{x})\psi^{+}(\vec{x}^{\prime})\frac{e^{2}}{|\vec{x}-\vec{x}^{\prime}|}\psi(\vec{x}^{\prime})\psi(\vec{x})+e\int
d\vec{x}V_{ext}(\vec{x},t)\psi^{+}(\vec{x})\psi(\vec{x})\enspace.$

Here according to the general recipe of second quantization a normal ordering
$N(...)$ had to be introduced also with respect to the photon creation and
annihilation operators $b_{\vec{q},\lambda}^{+}$, $b_{\vec{q},\lambda}$ and
the photon frequency is $\omega_{q}=c|q|$.

This non-relativistic QED Hamiltonian coincides with the standard one obtained
directly from the second quantized Hamilton operator of electrons interacting
with a classical electromagnetic field in the Coulomb gauge, after the
quantization of the transverse vector potential according to Eq. 39 and adding
the energy of the photons, as it is given for example in [7], while in most
papers external potentials are not included.

The interaction terms in the Hamiltonian Eq. 40 (in the absence of external
fields!) are besides the usual Coulomb one of the many-body theories, a
current-(transverse) field interaction $\frac{1}{c}\vec{j}{\vec{A}}_{\bot}$
and a ”sea-gull” term
$\frac{e^{2}}{2mc^{2}}\psi^{+}\psi{{\vec{A}}_{\bot}}^{2}$. The corresponding
vertexes of the Feynman diagram technique within the adiabatic perturbation
theory are illustrated on Fig. 1.

Figure 1: The Feynman vertexes of the non-relativistic QED. The dashed line is
the Coulomb potential, while the wavy line is the transverse photon.

Obviously, this quantization procedure reintroduces electromagnetic self-
interaction on a hidden way. Its manifestation is the much discussed self-
energy in the Feynman diagramms of the adiabatic perturbation theory. This
occurs also by the second-quantized version of the Coulomb interacting charged
particles. The Hamiltonian in configuration space is not identical with the
second quantized Coulomb Hamiltonian. Obviously their ground states (vacuum)
are different.

The Heisenberg equations of motion lead to the quantized (operator) coupled
Maxwell and Schrödinger equations. However their gauge invariant form does not
imply gauge invariance against arbitrary gauge transformation of the vector
potential operator $\vec{A}(\vec{x})$. Gauge transformations like Eq.18 are
meaningful only with c-number functions $\chi(\vec{x},t)$ for the external
field.

We have shown, how a field theoretical treatment allows the use of the
Lagrange formalism in deriving the non-relativistic quantum mechanical many-
body theory of charged particles interacting with photons avoiding the
problems linked to point-like classical charges. Of course these results may
be extended without problems to include also other charged particles, like the
positive ions or holes of the solid state theory.

##  7 Ingredients from the relativistic quantum field theory.

Until now we described the QED of non-relativistic particles characterized by
their mass and charge. However, from the relativistic quantum-field theory one
knows, that these are not the only basic properties of elementary or composite
particles. Although the non-relativistic QED is not reducible to the
relativistic QED, one must take into account some ingredients one has learned
from the quantum-field theory. At least the spin and the associated magnetic
moment of electrons and nuclei (or ions) are indispensable in a reasonable
condensed matter theory. An essential role plays in this context the spin-
statistic theorem of the relativistic theory stating that particles with
integer spin are bosons, while those with half integer spin are fermions.

For this sake one has to include in the classical Maxwell equations Eq.5 also
a magnetic (or spin ) current

$\vec{j}^{M}(\vec{x})=c\nabla\times\vec{M}(\vec{x})\enspace.$ (41)

The magnetization $\vec{M}(\vec{x})$ is given by the spin density. In the here
discussed example of electrons with spin $1/2$

$\vec{M}(\vec{x})=\frac{e\hbar}{2mc}\vec{\sigma}(\vec{x})\enspace,$ (42)

with the spin density

$\vec{\sigma}(\vec{x})=\sum_{\sigma,\sigma^{\prime}=\pm
1}\psi_{\sigma}(\vec{x})^{*}\hat{\vec{\sigma}}_{\sigma,\sigma^{\prime}}\psi_{\sigma^{\prime}}(\vec{x})$
(43)

and the well-known $2\times 2$ sigma matrices $\hat{\vec{\sigma}}$.

To get the so extended Maxwell equations one has to add the piece

$-\vec{B}(\vec{x})\vec{M}(\vec{x})$ (44)

to the Lagrangian density Eq.4 or Eq.4, respectively with the opposite sign to
the Hamiltonians Eq.5 and Eq.40. The wave functions $\psi_{\sigma}(\vec{x})$
will then satisfy the Pauli equation

$\displaystyle\imath\hbar\frac{\partial}{\partial t}\psi({\vec{x}},t)$
$\displaystyle=$
$\displaystyle\left(\frac{1}{2m}\left(-\imath\hbar\nabla+\frac{e}{c}(\vec{A}(x,t)+\vec{A}_{ext})\right)^{2}\right.$
$\displaystyle+$
$\displaystyle\left.e\left(V({\vec{x}},t)+V_{ext}(\vec{x},t)\right)+\frac{e\hbar}{2mc}\vec{B}(\vec{x})\hat{\vec{\sigma}}\right)\psi({\vec{x}},t)\enspace,$

with the matrix-column notation

$\psi(\vec{x},t)=\left(\begin{array}[]{cl}\psi_{+}(\vec{x})\\\
\psi_{-}(\vec{x})\end{array}\right)\enspace.$ (46)

The added piece to the Lagrangians does not affect gauge-invariance and the
quantization of this extended electromagnetic theory is trivial. Of course one
has to take into account the connection between the spin and statistics
(Fermi/Bose). Obviously, the scheme we described in these chapters may be
easily extended to any system of particles with arbitrary masses, charges and
spins.

##  8 Non-relativistic quantum-mechanical many-body Hamiltonian without
photons including $1\over c^{2}$ terms.

Only quantum optics uses up to these days the above described non-relativistic
QED of ensembles of charged particles and photons. Many-body theories of
solid-state or plasma restrict their task to the subspace of states containing
no photons. In the description of electrical phenomena with longitudinal
fields this theory hat enormous successes in understanding the properties of
solids. However, the description of magnetic properties is not so
consolidated, since the standard solid-state theory is still based just on
Coulomb forces if we leave aside spin magnetism.

The purpose of this Section is to derive from the non-relativistic QED the
proper formulation of the $1/c^{2}$ many-body theory containing already the
magnetic interactions between electrons. More than that is not possible
without inclusion of the photons.

To simplify the discussion we omit here the external fields and reintroduce
them again at the end according to the ”minimal rule” and denote by

$\vec{i}(\vec{x},t)=\frac{e}{2m}\psi(\vec{x},t)^{*}\frac{1}{\imath}\hbar\nabla\psi(\vec{x},t)+c.c$
(47)

the current density in the absence of a vector potential.

We shall proceed in two steps: i) to look in the subspace of states without
photons at the role of the different terms in the QED Hamiltonian we derived
and ii) to neglect retardation effects, that are not tractable with a local
Hamiltonian in this restricted subspace and anyway give rise to terms of
higher order as $1/c^{2}$.

First we discuss the theory in the absence of external fields. The terms that
do not contain the transverse vector potential ${\vec{A}}_{\bot}$ remain
unchanged. The term describing the energy of the free photons of course has to
be ignored. Then remains the Coulomb interaction and three other sort of
interaction terms we have to discuss. First at all the quadratic, so called
”sea-gull” term $\frac{e^{2}}{2mc^{2}}\psi^{+}\psi{{\vec{A}}_{\bot}}^{2}$. It
may be also ignored, since it communicates with the photon vacuum only with
the help of the terms $\frac{1}{c}\vec{i}{\vec{A}}_{\bot}$ and therefore may
contribute only third order terms in $1/c$. Thus we remain only with these
last interaction terms to discuss.

We shall pursue the discussion within the frame of the adiabatic $S$-matrix
theory (or of their Green functions). It is easy to see, that there is only
one basic graph and its combinations that may contribute by this vertex to the
$S$ matrix elements in the subspace of electron states. This is shown in Fig.
2. It describes the exchange of a transverse photon between two transverse
electron currents mediated by the transverse photon propagator given in the
4-dimensional Fourier space $\omega,{\vec{q}}$ by

$\frac{1}{q^{2}-\omega^{2}/c^{2}-\imath
0}(\delta_{\mu,\nu}-\frac{q_{\mu}q_{\nu}}{q^{2}})\quad;\qquad(\mu,\nu=1,2,3)$
(48)

and the vertexes contain momentum factors corresponding to the the currents
(see Ref. [6] )

After neglecting the term $-\omega^{2}/c^{2}$ in the denominator (i.e.
ignoring retardation) one eliminates corrections of higher order as $1/c^{2}$
and the photon propagator looks as

$\frac{1}{q^{2}}(\delta_{\mu,\nu}-\frac{q_{\mu}q_{\nu}}{q^{2}})\quad;\qquad(\mu,\nu=1,2,3)\enspace.$
(49)

Since no pole survived, the $-\imath 0$ term could have been also ignored and
$\frac{4\pi}{q^{2}}$ is just the Fourier transform of the Coulomb potential.

Figure 2: The transverse photon exchange diagram in the QED. The wavy line
represents the propagator of the transverse photon.

Then one may convince oneself that these graphs coincide with the basic
vertexes of the S matrix of an $1/c^{2}$ purely electron Hamiltonian
containing besides the kinetic energy and the Coulomb interactions the
interaction term

$-\frac{1}{2}\int d\vec{x}\int
d\vec{x^{\prime}}\frac{N\left[\vec{i}_{\bot}(\vec{x})\vec{i}_{\bot}(\vec{x}^{\prime})\right]}{c^{2}|\vec{x}-\vec{x}^{\prime}|}\enspace.$
(50)

This term has an appealing form analogous to the charge density-charge density
Coulomb interaction. It is the microscopical expression of the important Biot-
Savart law of interaction between currents in finite macroscopic samples,
where light propagation effects may be ignored.

One might argue that due to the smallness of the velocities in the condensed
matter such an $1/c^{2}$ term may be neglected. This is obviously false. Our
everyday experience teaches us, that a macroscopic number of slow electrons
may create enormous magnetic fields.

In order to keep gauge invariance with respect to an external field
$\vec{A}_{ext}$ one has to use again the minimal recipe and this requires also
its introduction not only in the kinetic energy term, but also in the current-
current interaction.

This transverse current-current interaction Eq. 50 was derived in Ref. [5],[6]
within this QED frame.

Therefore, we may conclude that after reintroducing the external fields
according to the minimal principle of Yang-Mills theories, the quantum
mechanical $1/c^{2}$ electron Hamiltonian that generates the diagrams of Fig.3

Figure 3: The basic density-density and current-current vertexes in the
$1/c^{2}$ S-matrix. Here the dashed line indicates just a Coulomb potential.

is just

$\displaystyle{\bf H}(t)\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\-\\!\\!\int\\!\\!d\vec{x}\psi^{+}(\vec{x})\left[\frac{1}{2m}\left(\frac{\hbar}{\imath}\nabla-\frac{e}{c}\vec{A}_{ext}(\vec{x},t)\right)^{2}\\!\\!+\\!\\!eV_{ext}(\vec{x},t)\right]\psi(\vec{x})$
$\displaystyle+$ $\displaystyle\frac{1}{2}\int d\vec{x}\int
d{\vec{x}}\,^{\prime}\frac{{\cal
N}\left[\rho(\vec{x})\rho({\vec{x}}\,^{\prime})\right]}{|\vec{x}-{\vec{x}}\,^{\prime}|}-\frac{1}{2}\int
d\vec{x}\int d{\vec{x}}\,^{\prime}\frac{{\cal
N}\left[\vec{j}_{\bot}(\vec{x},t)\vec{j}_{\bot}({\vec{x}}\,^{\prime},t)\right]}{c^{2}|\vec{x}-{\vec{x}}\,^{\prime}|}\enspace.$

Here $\rho(\vec{x})$ denotes the charge density operator

$\rho(\vec{x})=e\psi^{+}(\vec{x})\psi(\vec{x})\enspace,$ (52)

while $\vec{j}_{\bot}(\vec{x},t)$ denotes the transverse part of the current
density operator in the presence of the external vector potential

$\\!\vec{j}(\vec{x},t)\\!=\\!\frac{e}{2m}\left(\\!\psi^{+}(\vec{x})\left(\frac{\hbar}{\imath}\nabla\\!-\\!\frac{e}{c}\vec{A}_{ext}(\vec{x},t)\\!\\!\right)\psi(\vec{x})\\!+\\!h.c.\right)\enspace.$
(53)

The continuity equation emerging from this charge conserving Hamiltonian is
however

$\nabla\vec{J}+\frac{\partial}{\partial t}\rho=0$ (54)

with the conserved current

$\vec{J}(\vec{x},t)=\frac{e}{2m}\psi({\vec{x}},t)^{*}\left(-\imath\hbar\nabla+\frac{e}{c}({\vec{A}}_{ext}({\vec{x}},t)+{\vec{A}}_{\bot}(\vec{x},t))\right)\psi({\vec{x}},t)+c.c\enspace,$
(55)

where

$\vec{A}_{\bot}(\vec{x},t)\doteq\frac{1}{c}\int
d{\vec{x}}\,^{\prime}\frac{\vec{j}_{\bot}({\vec{x}}\,^{\prime},t)}{|\vec{x}-{\vec{x}}\,^{\prime}|}$
(56)

is (by neglecting the retardation!) the transverse vector potential generated
by the electrons.

The generalization of these results for a system of electrons and ions as
constituents of the solid state is obvious.

This new interaction in coordinate space, due to the additional integrals in
the definition of the transverse part and the normal products, is difficult to
handle. However, this term has a simple expression in the discrete ${\vec{k}}$
\- space basis (plane waves with periodical boundary conditions in a cube of
volume $\Omega$). It looks explicitly as

$-\frac{e^{2}\hbar^{2}}{m^{2}c^{2}\Omega}\sum_{\sigma,\sigma^{\prime}=\pm
1}\sum_{\vec{k},\vec{p},\vec{q}}\frac{2\pi}{q^{2}}\left(\vec{k}\vec{p}-\vec{q}\vec{k}\frac{1}{q^{2}}\vec{q}\vec{p}\right)a_{{\vec{k}},\sigma}^{+}a_{\vec{p},\sigma^{\prime}}^{+}a_{\vec{p}+\vec{q},\sigma^{\prime}}a_{\vec{k}-\vec{q},\sigma}\enspace.$
(57)

For a better understanding of the underlying physics, let us consider now the
Hartree approximation of this Hamiltonian. It looks as

$\displaystyle{\bf H}_{Hartree}(t)\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\-\\!\\!\int\\!\\!d{\vec{x}}\psi^{+}(\vec{x})\left[\frac{1}{2m}\left(\frac{\hbar}{\imath}\nabla-\frac{e}{c}\vec{A}_{ext}(\vec{x},t)\right)^{2}\\!\\!+\\!\\!eV_{ext}(\vec{x},t)\right]\psi(\vec{x})$
$\displaystyle+$ $\displaystyle\int d\vec{x}\int
d{\vec{x}}\,^{\prime}\frac{\rho(\vec{x})\langle\rho({\vec{x}}\,^{\prime},t)\rangle}{|\vec{x}-{\vec{x}}\,^{\prime}|}-\int
d\vec{x}\int
d{\vec{x}}\,^{\prime}\frac{\vec{j}_{\bot}(\vec{x},t)\langle\vec{j}_{\bot}({\vec{x}}\,^{\prime},t)\rangle}{c^{2}|\vec{x}-{\vec{x}}\,^{\prime}|}\enspace,$

where $\langle\rho(\vec{x},t)\rangle$ and
$\langle{\vec{j}}_{\bot}(\vec{x},t)\rangle$ are the (chosen) ensemble averages
of the charge and transverse current densities.

One may here identify the s.c. internal scalar and vector potentials

$V_{int}(\vec{x},t)=\int
d{\vec{x}}\,^{\prime}\frac{\langle\rho({\vec{x}}\,^{\prime},t)\rangle}{|\vec{x}-{\vec{x}}\,^{\prime}|};\qquad\vec{A}_{int}(\vec{x},t)=\int
d{\vec{x}}\,^{\prime}\frac{\langle
j_{\bot}({\vec{x}}\,^{\prime},t)\rangle}{c|\vec{x}-{\vec{x}}\,^{\prime}|}\enspace.$
(59)

Of course, here in the definition of the internal vector potential the
retardation is again missing!

Eq.8 and Eq.8 together with the identifications of Eq.59 show, that the
average Coulomb field created by the electron results from the charge-charge
interaction and the average (dia-) magnetic field created by the velocity of
the electron results from the current-current interactions. This is analogous
to the (ferro-) magnetic field of localized spins resulting from their mutual
interaction in the Heisenberg spin model.

One may rewrite Eq. 8 as

$\displaystyle{\bf H}_{Hartree}(t)$ $\displaystyle=$ $\displaystyle-\int
d{\vec{x}}\psi^{+}(\vec{x})\left[\frac{1}{2m}\left(\frac{\hbar}{\imath}\nabla-\frac{e}{c}\vec{A}_{ext}(\vec{x},t)\right)^{2}+eV_{tot}(\vec{x},t)\right]\psi(\vec{x})$
(60) $\displaystyle-$ $\displaystyle\frac{1}{c}\\!\int
d\vec{x}\int\vec{j}_{\bot}(\vec{x},t)\vec{A}_{int}(\vec{x},t)\enspace.$

The above Hartree Hamiltonian coincides only up to second order terms in the
fields with the mean field Hamiltonian resulted from the QED Hamiltonian Eq.40
by approximating the total vector potential
${\vec{A}}_{\bot}({\vec{x}})+{\vec{A}}_{ext}$ by his average $\vec{A}_{tot}$
and the Coulomb term by its mean-field approximation.

$H_{mean}=\int
d{\vec{x}}\psi^{+}(\vec{x})\left[\frac{1}{2m}\left(\frac{\hbar}{\imath}\nabla-\frac{e}{c}\vec{A}_{tot}(\vec{x},t)\right)^{2}\\!\\!+\\!\\!eV_{tot}(\vec{x},t)\right]\psi(\vec{x})\enspace.$
(61)

While the Hamiltonian Eq.60 is invariant only against time-independent gauge
transformations of the external fields, the mean-field Hamiltonian Eq. 61 is
formally gauge invariant against time independent gauge transformations of the
total field $\vec{A}_{tot}$ although actually the gauge of the internal vector
potential ${\vec{A}}_{int}$ is already fixed to be the Coulomb one.

##  9 Darwin’s classical approach revisited.

In this Section we show that Darwin’s reasoning within the classical
electromagnetic theory of point-like charges, if properly formulated in the
Coulomb gauge i.e. considering only the true physical degrees of freedom of
the magnetic field, leads to a classical Hamiltonian analogous to the one we
got in the previous Section.

As it is well-known, one can not formulate a Lagrangian theory of classical
point-like charged particles interacting with the electromagnetic field due to
the divergent self-interaction.( From the Lorentz force one has to omit the
action of the field created by each charged particle on itself!) This impedes
also a proper derivation of an $1/c^{2}$ Hamiltonian.

Almost one hundred years ago Darwin [1] proposed nevertheless a closed
classical Lagrangian for N point-like charges $e_{i}$ and mass $m_{i}$
($i,j=1,\ldots N$) including terms up to order $1/c^{2}$ avoiding self-
interaction and constructed the following Hamiltonian

$\displaystyle{\cal
H}=\sum_{i}\frac{1}{2m_{i}}\vec{p_{i}}^{2}+\sum_{i>j}\frac{e_{i}e_{j}}{|\vec{r_{i}}-\vec{r}_{j}|}$
(62)
$\displaystyle-\sum_{i>j}\frac{e_{i}e_{j}}{2c^{2}m_{i}m_{j}|\vec{r}_{i}-\vec{r}_{j}|}\left[\vec{p}_{i}\cdot\vec{p}_{j}+(\vec{p}_{i}\cdot\vec{n}_{ij})(\vec{p}_{j}\cdot\vec{n}_{ij})\right],$

where
$\vec{n}_{ij}\equiv\frac{\vec{r}_{i}-\vec{r}_{j}}{|\vec{r}_{i}-\vec{r}_{j}|}$.
His derivation is based on the expansion of the Liénard-Wiechert potentials to
second order in $\frac{1}{c}$.

The reasoning of Refs. [2] and [3] to derive the above Hamiltonian starts from
the discussion of the potentials felt by one point-like particle in the field
of another one at a distance $R$. Landau and Lifshitz [2] use the Lorentz
gauge, perform an expansion of the scalar potential up to second order in the
finite distance between two particles divided by the light velocity ($R/c$)
and later perform a gauge transformation turning to the Coulomb gauge, without
stating it explicitly.

Jackson [3] starts in the Coulomb gauge, gets the transverse current -
transverse current Lagrange function for two particles however performs an
integration by parts of the vector potential of the two particles at a finite
distance leaving aside the vanishing surface contribution and gets Darwin’s
Hamiltonian.

Thus the two approaches are similar and their results are identical. Moreover,
from Jackson’s derivation results that the transverse current - transverse
current version is equivalent to Darwin’s version whenever the distances
between the particles are finite.

Without the mentioned partial integration Jackson’s Hamiltonian [3] looks as

$\displaystyle
H=\sum_{i}\frac{\vec{p_{i}}^{2}}{2m_{i}}+\sum_{i>j}\frac{e_{i}e_{j}}{|\vec{r_{i}}-\vec{r}_{j}|}$
(63)
$\displaystyle-\sum_{i>j}\frac{e_{i}e_{j}}{c^{2}m_{i}m_{j}}\vec{p}_{i}\left[\frac{\vec{p}_{j}}{|\vec{r}_{i}-\vec{r}_{j}|}-\frac{1}{4\pi}\int
d\vec{x}\frac{1}{|\vec{r}_{i}-\vec{x}|}\nabla\left(\vec{p}_{j}\nabla\frac{1}{|\vec{x}-\vec{r}_{j}|}\right)\right]$

and consists of transverse current - transverse current terms for $i\neq j$.

By introducing the charge and current densities:

$\rho(\vec{x})=\sum_{i}e_{i}\delta(\vec{x}-\vec{r}_{i});\qquad\vec{i}(\vec{x})=\sum_{i}\frac{e_{i}}{m_{i}}\vec{p}_{i}\delta(\vec{x}-\vec{r}_{i})$
(64)

and ignoring the missing $i=j$ terms one might rewrite this classical
Hamiltonia Eq.63 as

$\sum_{i}\frac{1}{2m_{i}}\vec{p_{i}}^{2}+\frac{1}{2}\int d\vec{x}\int
d\vec{x}^{\prime}\frac{\rho(\vec{x})\rho(\vec{x}^{\prime})}{|\vec{x}-\vec{x}^{\prime})|}-\frac{1}{2}\int
d\vec{x}\int
d\vec{x^{\prime}}\frac{\vec{i}_{\bot}(\vec{x})\vec{i}_{\bot}(\vec{x}^{\prime})}{c^{2}|\vec{x}-\vec{x}^{\prime}|}\enspace,$
(65)

where $\vec{i}_{\bot}(\vec{x})$ is the transverse part of the current density

$\vec{i}_{\bot}(\vec{r},t)\equiv\vec{i}(\vec{r},t)+\frac{1}{4\pi}\nabla\int
d\vec{r}^{\prime}\frac{\nabla^{\prime}\vec{i}(\vec{r}^{\prime},t)}{|\vec{r}-\vec{r}^{\prime}|}\enspace.$
(66)

Under this form it looks similar to the result we obtained before (in the
absence of other external fields). A quantization of the classical Hamiltonian
of the point-like charged particles Eq. 63 is then immediate by second
quantization of Eq. 65 as it was done in Refs. [5], [6] avoiding the
configuration space quantization either of Eq. 63 or Eq. 62. This leads to the
same Hamiltonian Eq. 8 as we deduced from the non-relativistic QED.

##  10 The importance of the diamagnetic current-current interaction in many-
body theories.

###  10.1 Superconductivity

According to Bardeen, Cooper and Schrieffer [11] the origin of the
superconductive phase transition lies in the correlation between electrons of
opposite momenta and spin resulted from phonon exchange. However, to pursue
this idea within the many-body theory of electrons interacting with phonons
seems too difficult. Therefore, one tried to construct pure electron many-body
theories with a built-in two-particle”attractive” potential $W(\vec{x})$
giving rise to no bound states, but just to such correlations and implicitly
to a superconducting phase transition. Such a model Hamiltonian is due to
Rickayzen [12, 13] that we shortly describe here. (The specific version of
Bogolyubov-de Gennes [14] is included in the same frame.) The superconductive
phase transition is thought to be accompanied by a spontaneous breaking of the
phase invariance due to the non-vanishing anomalous average
$\langle\psi_{\frac{1}{2}}^{+}(\vec{x})\psi_{-\frac{1}{2}}^{+}(\vec{x^{\prime}})\rangle$
.

Rickayzen’s self-consistent Hamiltonian for a homogeneous electron system in
the presence of a magnetic field described by the potential vector ${\vec{\cal
A}}({\vec{x}})$ is

$\displaystyle{\cal H}_{s.c.}=\sum_{\sigma=\pm\frac{1}{2}}\\!\\!\\!\int
d\vec{x}\psi_{\sigma}^{+}(\vec{x})\\!\bigg{\\{}\frac{1}{2m}\left(-\imath\hbar\nabla-\frac{e}{c}\vec{\cal{A}}(\vec{x})\right)^{2}-\mu\bigg{\\}}\psi_{\sigma}(\vec{x})$
(67) $\displaystyle+\frac{1}{2}\int d\vec{x}\int
d{\vec{x}}\,^{\prime}W(\vec{x}-{\vec{x}}\,^{\prime})\left[\langle\psi_{\frac{1}{2}}^{+}(\vec{x})\psi_{-\frac{1}{2}}^{+}({\vec{x}}\,^{\prime})\rangle\psi_{-\frac{1}{2}}({\vec{x}}\,^{\prime})\psi_{\frac{1}{2}}(\vec{x})\right.$
$\displaystyle+\left.\langle\psi_{-\frac{1}{2}}({\vec{x}}\,^{\prime})\psi_{\frac{1}{2}}(\vec{x})\rangle\psi_{\frac{1}{2}}^{+}(\vec{x})\psi_{-\frac{1}{2}}^{+}({\vec{x}}\,^{\prime})-\langle\psi_{\frac{1}{2}}^{+}(\vec{x})\psi_{-\frac{1}{2}}^{+}({\vec{x}}\,^{\prime})\rangle\langle\psi_{-\frac{1}{2}}({\vec{x}}\,^{\prime})\psi_{\frac{1}{2}}(\vec{x})\rangle\right]\enspace.$

One can show that in the absence of the field ${\vec{\cal A}}({\vec{x}})$ a
phase transition may occur below a critical temperature, provided the
potential $W({\vec{x}})$ ensures a non-vanishing solution for the symmetry
breaking gap parameter of the largely described ”gap equation” we do not give
here. This condition is equivalent to the vanishing of the first derivative of
the free energy with respect to the same parameter.

In order to prove the ideal diamagnetism (Meissner effect) one uses
equilibrium linear response. Here it is essential to understand the magnetic
perturbation. In the Hamiltonian of Eq.67 the (transverse) vector potential
$\vec{\cal{A}}$ is a classical field that was introduced just by the minimal
principle, without any deeper justification.

According to our previous Sections (see Eqs. 8 and 59 ) the correct s.c.
Hamiltonian [16] derived from the QED in the absence of photons is quite
different, namely the kinetic energy term of Eq. 67 in the presence of a
transverse external vector potential is replaced by

$\psi_{\sigma}^{+}(\vec{x})\\!\frac{1}{2m}\left(-\imath\hbar\nabla-\frac{e}{c}\vec{A}_{ext}(\vec{x})\right)^{2}\psi_{\sigma}(\vec{x})-\frac{1}{c}\vec{A}_{int}(\vec{x})\vec{j}_{\bot}(\vec{x})\enspace.$
(68)

In the linear approximation with respect to $\vec{A}_{ext}$ one gets the
magnetic perturbing term

$-\frac{1}{c}\big{(}\vec{A}_{ext}(\vec{x})+\vec{A}_{int}(\vec{x})\big{)}\vec{j}_{\bot}(\vec{x})=-\frac{1}{c}\vec{A}_{tot}(\vec{x})\vec{j}_{\bot}(\vec{x})\enspace.$
(69)

Therefore, at least at the level of the linear response one should identify
the vector potential $\vec{\cal{A}}$ not with the external but with total one.

Taking also into account the peculiarity of the linear response within self-
consistent theories that the deviation of the averages from their equilibrium
values constitute so called induced perturbations one may derive the relation
between the average induced current and this total field (in Fourier
transforms) of the two transverse vectors reads as

$\langle\tilde{j}_{\mu}({\vec{k}})\rangle=\kappa(k)\tilde{{\cal
A}}_{\mu}({\vec{k}})\quad;\qquad(\mu=1,2,3)$ (70)

with the scalar coefficient $\kappa(k)$ in an infinite homogeneous, isotropic
system being a function of $k=\sqrt{{{\vec{k}}}^{2}}$. Although this
coefficient cannot be calculated explicitly, it has been proven [17] that for
any potential $W(\vec{x})$ that leads to a stable superconductive phase in the
absence of a magnetic field the coefficient $\kappa(0)$ is finite. This was
early stipulated by Schafroth [18] as being the necessary condition for
superconductivity.

Thus we may conclude that

$\langle\vec{\tilde{j}}({\vec{k}})\rangle=\kappa(k)\big{(}\vec{\tilde{A}}_{ext}({\vec{k}})+\vec{\tilde{A}}_{int}({\vec{k}})\big{)}\quad;\qquad(\mu=1,2,3)\enspace.$
(71)

Now, in the absence of retardation we have the relation of the internal vector
potential to the average transverse current density:

$\vec{\tilde{A}}_{int}({\vec{k}})=\frac{4\pi}{ck^{2}}\langle\vec{\tilde{j}}_{\bot}({\vec{k}})\rangle\enspace.$
(72)

Using this relationship in the previous equation we get

$\tilde{A}_{int}^{\mu}({\vec{k}})=\frac{\frac{4\pi}{ck^{2}}\kappa(k)}{1-\frac{4\pi}{ck^{2}}\kappa(k)}\tilde{A}_{ext}^{\mu}({\vec{k}})\quad;\qquad(\mu=1,2,3)\enspace.$
(73)

For the internal and external magnetic fields holds similarly

${\tilde{B}}^{\mu}_{int}({\vec{k}})=\frac{\frac{4\pi}{ck^{2}}\kappa(k)}{1-\frac{4\pi}{ck^{2}}\kappa(k)}{\tilde{B}}^{\mu}_{ext}({\vec{k}})\quad;\qquad(\mu=1,2,3)\enspace.$
(74)

This last equation has been obtained also by Tinkham [19] assuming the
interpretation of the vector potential $\vec{\cal{A}}$ in Eq. 8 as being the
total vector potential as in Eq. 61. Although it was just an educated guess,
it occurred to be correct within the linear approximation. The same is true
about the Ginzburg-Landau non-linear theory of superconductivity [15].

The same relation Eq. 74 may be obtained through Zubarev’s [8] early reasoning
based just on the macroscopic Maxwell equations, without any reference to a
Hamiltonian.

Since $\kappa(0)\neq 0$, from Eq. 74 follows

${\tilde{B}}^{\mu}_{int}(0)=-{\tilde{B}}^{\mu}_{ext}(0)\enspace$ (75)

i.e. a homogeneous external magnetic field is perfectly compensated by the
internal magnetic field of the electrons. This shows that at least against a
homogeneous magnetic field the system behaves as an ideal diamagnet.

As we have seen, although the correct electromagnetic formulation at the level
of the linear response did not predict unexpected results, it eliminated some
conceptual confusions. On the other hand, by strong magnetic fields, where
linear response does not work, the correct $1/c^{2}$ Hamiltonian Eq.8 may be
very important.

###  10.2 Linear response and thermal noise

####  10.2.1 Linear response of charged particles.

Non-equilibrium linear response and the related problem of thermal
fluctuations is another field where the proper treatment of the magnetic field
within many-body theory is important.

Originally Kubo [20] developed adiabatic linear response only with respect to
a homogeneous electric field. However, it may be extended within the non-
relativistic QED to a general external electromagnetic perturbation due to
applied electromagnetic scalar $V^{ext}(\vec{x},t)$ and vector potentials
$\vec{A}^{ext}(\vec{x},t)$

$H^{\prime}(t)=\int
d\vec{x}\left\\{\rho(\vec{x})V^{ext}(\vec{x},t)-\vec{j}(\vec{x})\vec{A}^{ext}(\vec{x},t)\right\\}\enspace,$
(76)

the average of the current density operator $\vec{j}(\vec{x},t)$ being given
by

$\langle
j_{\mu}(\vec{x},t)\rangle\\!=\\!\\!\\!\lim_{s\to+0}\int_{-\infty}^{t}\\!\\!\\!\\!\\!dt^{\prime}e^{st^{\prime}}\\!\int_{0}^{\beta}\\!\\!\\!\\!d\lambda\\!\int\\!\\!\\!d{\vec{x}}\,^{\prime}\langle
j_{\nu}({\vec{x}}\,^{\prime},-\imath\hbar\lambda)j_{\mu}(\vec{x},t-t^{\prime})\rangle_{0}E_{\nu}^{ext}({\vec{x}}\,^{\prime},t^{\prime})\enspace,$
(77)

where $\mu,\nu=1,2,3$ are vector-indices and summation over double indices is
understood, while $s$ is the adiabatic parameter we shall omit for simplicity
in the following.

In translation and rotation (in space) invariant systems one has in Fourier
transforms a local relationship

$\langle{\tilde{j}}_{\mu}({\vec{k}},\omega)\rangle=\kappa({\vec{k}},\omega)_{\mu\nu}\tilde{E}^{ext}_{\nu}(\omega,{\vec{k}})$
(78)

with

$\kappa({\vec{k}},\omega)_{\mu\nu}=\int_{0}^{\infty}\\!\\!\\!dt\int_{0}^{\beta}\\!\\!\\!d\lambda\int\\!\\!\\!d\vec{x}e^{\imath({\vec{k}}\vec{x}+\omega
t)}\langle
j_{\nu}(0,0)j_{\mu}(\vec{x},t+\imath\hbar\lambda)\rangle_{0}\enspace.$ (79)

If the system under consideration is isotropic, then one may separate the
longitudinal and transverse parts

$\kappa({\vec{k}},\omega)_{\mu\nu}=\frac{k_{\mu}k_{\nu}}{k^{2}}\kappa_{L}(k,\omega)+\left(\delta_{\mu\nu}-\frac{k_{\mu}k_{\nu}}{k^{2}}\right)\kappa_{T}(k,\omega)\enspace.$
(80)

It is important to remark here that the frequency and the wave vector in the
transverse case are not independent ($\omega=ck$).

The next important point, first remarked by Izuyama [21] and [8] is that
$\kappa({\vec{k}})$ does not coincide with the complex conductivity
$\sigma(\omega,{\vec{k}})$, since this one characterizes the response to the
total electric field in matter to which the electrons also contribute. This
total electric field is given by adding the average internal field to the
external field

$\vec{E}({\vec{k}},t)=\vec{E}^{ext}({\vec{k}},t)+\vec{E}^{int}({\vec{k}},t)\enspace.$
(81)

According to the Maxwell equations of the non-relativistic QED the Fourier
transforms of the internal longitudinal and transverse electric fields are
given by

$\imath{\vec{k}}\vec{\tilde{E}}_{L}({\vec{k}},t)=\frac{4\pi}{k}\tilde{\rho}({\vec{k}},t)$
(82)

and

$\vec{E}_{T}(\vec{x},t)=-\frac{1}{c}\frac{\partial}{\partial
t}\vec{A}_{T}(\vec{x},t)\enspace,$ (83)

with the transverse vector potential operator (radiation field) given by Eq.
39. After neglecting retardation the transverse electric field may be
expressed directly through the transverse current density

$\vec{\tilde{E}}_{T}({\vec{k}},t)=-\frac{4\pi}{(ck)^{2}}\frac{\partial}{\partial
t}\vec{\tilde{j}}_{T}({\vec{k}},t)\enspace.$ (84)

The internal fields are then defined by the average charge , respectively
current densities

$\displaystyle\imath{\vec{k}}\vec{\tilde{E}}_{L}^{int}({\vec{k}},t)$
$\displaystyle=$
$\displaystyle\frac{4\pi}{k}\langle\tilde{\rho}({\vec{k}},t)\rangle$ (85)
$\displaystyle\vec{\tilde{E}}_{T}^{int}({\vec{k}},t)$ $\displaystyle=$
$\displaystyle-\frac{4\pi}{(ck)^{2}}\frac{\partial}{\partial
t}\langle\vec{\tilde{j}}_{T}({\vec{k}},t)\rangle\enspace.$ (86)

As a consequence the longitudinal and transverse conductivities are related to
the coefficients $\kappa_{L/T}(k,\omega)$ (see [24]) by

$\sigma_{L}(k,\omega)=\frac{\kappa_{L}(k,\omega)}{1-\imath\frac{4\pi}{\omega}\kappa_{L}(k,\omega)}\enspace,$
(87)

respectively

$\sigma_{T}(k,\omega)=\frac{\kappa_{T}(k,\omega)}{1+\imath\frac{4\pi}{c^{2}k^{2}}\kappa_{T}(k,\omega)}\enspace.$
(88)

Since the transverse external electric field obeys the wave equation, in the
last equation the variables $\omega$ and $k$ are related by $\omega=ck$ !
Therefore one might speak only about a frequency dependent transverse
conductivity

$\sigma_{T}(\omega)=\frac{\kappa_{T}(\omega/c,\omega)}{1+\imath\frac{4\pi}{\omega}\kappa_{T}(\omega/c,\omega)}\enspace.$
(89)

In Eq. 84 relating the internal transverse electric field to the average
transverse electronic current we already neglected the retardation, therefore
it is consequent to neglect it also in calculating the coefficient
$\kappa_{T}(k,,\omega)$ within the $1/c^{2}$ Hamiltonian of Eq. 8 (including
possible other terms due to interaction with phonons or impurities).

####  10.2.2 Electromagnetic thermal noise.

There is an interesting relationship between these conductivities given by the
consequent linear response theory of charged particles and their thermal
noise. Such a connection was first shown in a fluctuation-dissipation theorem
by Callen and Welton [22]. However, in their derivation the electromagnetic
field of the charged particles was ignored. A proper electromagnetic analysis
for Coulomb interacting particles was performed in Ref. [23] and recently [24]
extended to include the transverse case with diamagnetic interaction.

The time fluctuation $\Delta_{X}(t)$ of a given observable (hermitian
operator) $X$ in thermal equilibrium is defined [25] as the average square
deviation

$\Delta_{X}(t)=\langle\left(X(t)-X(0)\right)^{2}\rangle_{0}\geq 0\enspace,$
(90)

where the average is taken over the macro-canonical equilibrium density
matrix. This is analogous to the general definition of a fluctuation both in
classical- or quantum-statistics.

Leaving apart the constant $2\langle X(0)^{2}\rangle$ usually chosen to be
vanishing, the entity of interest is

$\delta_{X}(t)=\langle X(t)X(0)+X(0)X(t)\rangle_{0}\enspace,$ (91)

which is a real and even function of $t$. As a consequence its Fourier
transform

${\tilde{\delta}_{X}}(\omega)=\int_{-\infty}^{\infty}dte^{\imath\omega
t}\delta_{X}(t)=2\int_{0}^{\infty}dt\cos(\omega t)\delta_{X}(t)\enspace$ (92)

is also real and even. Moreover, according to the Wiener-Khinchin Theorem [26,
27], [28] it is positive. This is defined [25] as the ”noise” spectrum of $X$
in quantum statistics

It is easy to show, by expansion in the basis of the eigenfunctions of the
Hamiltonian for any observable $X$ (here $X\equiv X(0)$) the following
identity

${\tilde{\delta}_{X}}(\omega)=2\hbar\omega\coth(\frac{\beta\hbar\omega}{2})\Re\int_{0}^{\infty}dte^{-\imath\omega
t}\int_{0}^{\beta}d\lambda\langle XX(t+\imath\hbar\lambda\rangle_{0}\enspace.$
(93)

As we have seen, this kind of correlators appear in the linear response
theory.

Now, let us consider the noise spectrum of an electric field. In Fourier
transforms it is easy to define the scalars characterizing the longitudinal,
respectively its transverse components
$E_{L}({\vec{k}})\equiv\frac{{\vec{k}}}{k}\vec{\tilde{E}}({\vec{k}},t)$ and
$E_{T(\lambda)}\equiv\vec{e_{\lambda}}\vec{\tilde{E}}({\vec{k}},t)$ with
$\vec{e}^{(\lambda)}_{\vec{k}}=\vec{e}^{(\lambda)}_{-{\vec{k}}}$
($\lambda=1,2$) being the two unit vectors defining the independent
polarizations of the transverse field.

From the hermiticity of the operator $\vec{E}(\vec{x},t)$ it follows that its
Fourier transform obeys
$\vec{\tilde{E}}(-{\vec{k}},t)^{+}=\vec{\tilde{E}}({\vec{k}},t)$. We may
define however two hermitian scalar operators (observables) as the ”real” and
”imaginary” parts of the operators. In what follows we shall omit the index
$\lambda$ since due to isotropy neither the proof nor the results depend and
consider the noise of the observables

$\displaystyle E_{L}^{R}({\vec{k}},t)$ $\displaystyle=$
$\displaystyle\frac{{\vec{k}}}{2k}(\vec{{\tilde{E}}}({\vec{k}},t)+\vec{{\tilde{E}}}(-{\vec{k}},t))$
(94) $\displaystyle E_{L}^{I}({\vec{k}},t)$ $\displaystyle=$
$\displaystyle\frac{{\vec{k}}}{2k\imath}(\vec{{\tilde{E}}}({\vec{k}},t)-\vec{{\tilde{E}}}(-{\vec{k}},t))\enspace,$
(95)

respectively

$\displaystyle E_{T}^{R}({\vec{k}},t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\vec{e}_{\vec{k}}(\vec{\tilde{E}}_{T}({\vec{k}},t)+\vec{\tilde{E}}_{T}(-{\vec{k}},t))$
(96) $\displaystyle E_{T}^{I}({\vec{k}},t)$ $\displaystyle=$
$\displaystyle\imath\frac{1}{2\imath}\vec{e}_{\vec{k}}(\vec{\tilde{E}}_{T}({\vec{k}},t)-\vec{\tilde{E}}_{T}(-{\vec{k}},t))\enspace.$
(97)

Then we use Eqs. 82, 84 that express these operators through the charge
density and current density operators while neglecting the retardation and
consequently considering the time evolution also in the $1/c^{2}$
approximation i.e. in the frame of the Hamiltonian Eq. 8.

Starting from definition of the above defined hermitian field operators and of
the noise spectral density Eq. 92, after some algebra using translation and
rotation invariance in the coordinate space, as well as some partial
integrations one gets for the longitudinal and transverse noise spectra
relationships to the complex conductivities defined by Eqs. 79, 87, 89

$\displaystyle{\tilde{\delta}}_{E_{L}}(\omega,k)$ $\displaystyle=$
$\displaystyle-8\pi\Omega\hbar\coth(\frac{\beta\hbar\omega}{2})\Im{\frac{1}{1+\frac{4\pi\imath}{\omega}\sigma_{L}(k,\omega)}}$
(98) $\displaystyle=$
$\displaystyle-8\pi\Omega\hbar\coth(\frac{\beta\hbar\omega}{2})\Im{\frac{1}{\epsilon_{L}(k,\omega)}}\enspace,$

respectively

${\tilde{\delta}}_{E_{T}}(\omega,k)|_{k=\omega/c}=8\pi\Omega\hbar\coth(\frac{\beta\hbar\omega}{2})\Im\frac{1}{1-\frac{4\pi\imath}{\omega}\sigma_{T}(\omega)}$
(99)

In Eq.98 we used the relation between the complex dielectric function and the
complex conductivity

$\epsilon(k,\omega)=1+\frac{4\pi\imath}{\omega}\sigma(k,\omega)$ (100)

and the inverted order of arguments $(\omega,k)$ in the notation underlines
that ${\vec{k}}$ is the wave-vector of the electric field, while $\omega$ is
the frequency of the noise spectrum.

These are the most general results for the noise spectrum of the electric
field of wave vector ${\vec{k}}$. From Eqs. 98, 99 and the positivity of the
noise spectral density follows also the positivity of the real part of the
longitudinal/transversal conductivity ($\Re\sigma(\omega,k)\geq 0$) within the
frame of the linear response theory.

In the $\hbar\to 0$ limit one gets the results of the classical plasma theory
[29].

In the peculiar case of a homogeneous in space longitudinal electric field
${\cal E}(t)$ (having a Fourier transform in space proportional to the volume
$\Omega=LS$) one gets for the potential drop $U(t)={\cal E}(t)L$ the noise
spectral density

${\tilde{\delta}}_{U}(\omega,0)=-8\pi\hbar\frac{L}{S}\coth(\frac{\beta\hbar\omega}{2})\Im{\frac{1}{1+\frac{4\pi\imath}{\omega}\sigma_{L}(\omega)}}$
(101)

Now, one has the frequency dependent resistance

$R(\omega)=\frac{L}{S\Re\sigma_{L}(\omega)}$ (102)

along the $z$ direction and one may define a capacity between the end cross-
sections of the sample

$C(\omega)=\frac{S}{L4\pi}\Re\epsilon_{L}(\omega)=\frac{S}{L4\pi}(1-\frac{4\pi}{\omega}\Im\sigma_{L}(\omega))\enspace.$
(103)

If the resistance and the capacitor are parallel linked, then the resulting
impedance $Z(\omega)$ is

$\frac{1}{Z(\omega)}=\frac{1}{R(\omega)}+\imath\omega C(\omega)$ (104)

then one gets from Eq.101

${\tilde{\delta}}_{U}(\omega,0)=2\hbar\omega\coth(\frac{\beta\hbar\omega}{2})\Re
Z(\omega)=2\hbar\omega\coth(\frac{\beta\hbar\omega}{2})\frac{R(\omega)}{1+(\omega
R(\omega)C(\omega))^{2}}\enspace.$ (105)

For vanishing noise frequency

${\tilde{\delta}}_{U}(0,0)=4Rk_{B}T$ (106)

and one recovers the old Nyquist theorem [30] .

Beside the field noises one might consider also the noise of the photon
occupation numbers. The noise spectral density of the photon occupation
numbers $n_{{\vec{k}}}$ (with an arbitrary chosen polarization not mentioned)
is given by

$\tilde{\delta}_{n_{{\vec{k}}}}(\omega)=2\hbar\omega\coth(\frac{\beta\hbar\omega}{2})\Re\int_{0}^{\infty}dte^{-\imath\omega
t}\int_{0}^{\beta}d\lambda\langle{\hat{n}}_{{\vec{k}}}{\hat{n}}_{{\vec{k}}}(t+\imath\hbar\lambda\rangle_{0}\enspace,$
(107)

where ${\hat{n}}_{{\vec{k}}}\equiv b_{{\vec{k}}}^{+}b_{{\vec{k}}}$ is the
operator of the number of photons of wave vector ${\vec{k}}$ and the given
polarization we do not mention in the notation.

On the other hand, by two partial integrations (using time translation
invariance and the assumed decay of the correlations) one may show that

$\int_{0}^{\infty}\\!\\!\\!\\!\\!dte^{-\imath\omega
t}\langle\hat{n}_{{\vec{k}}}(0)\hat{n}_{{\vec{k}}}(t+\imath\hbar\lambda)\rangle_{0}=\frac{1}{\omega^{2}}\int_{0}^{\infty}\\!\\!\\!\\!\\!dte^{-\imath\omega
t}\langle\dot{\hat{n}}_{{\vec{k}}}(0)\dot{\hat{n}}_{{\vec{k}}}(t+\imath\hbar\lambda)\rangle_{0}\enspace.$
(108)

However, retaining the lowest order in $1\over c$ from the full QED
Hamiltonian

$\frac{1}{c}\vec{j}_{T}(\vec{x},t)\vec{A}_{T}$ (109)

we have

${\dot{n}}_{{\vec{k}}}(t)=\frac{\imath}{\hbar
c}\int\\!\\!d\vec{x}\big{[}\vec{j}_{T}(\vec{x},t)\vec{A}_{T}(\vec{x},t),n_{{\vec{k}}}(t)\big{]}$
(110)

and

$\displaystyle\int_{0}^{\infty}\\!\\!\\!\\!\\!dte^{-\imath\omega t}\langle
n_{{\vec{k}}}(0)n_{{\vec{k}}}(t+\imath\hbar\lambda)\rangle_{0}=-\frac{1}{(\hbar\omega
c)^{2}}\int_{0}^{\infty}\\!\\!\\!\\!\\!dte^{-\imath\omega
t}\int\\!\\!d\vec{x}\int\\!\\!d{\vec{x}}\,^{\prime}$ (111)
$\displaystyle\langle\big{[}\vec{j}_{T}(\vec{x},0)\vec{A}_{T}(\vec{x},0),n_{{\vec{k}}}(0)\big{]}\big{[}\vec{j}_{T}({\vec{x}}\,^{\prime},t+\imath\hbar\lambda)\vec{A}_{T}({\vec{x}}\,^{\prime},t+\imath\hbar\lambda),n_{{\vec{k}}}(t+\imath\hbar\lambda)\big{]}\rangle_{0}\enspace,$

where the transverse vector potential operator is defined by Eq. 39.

In what follows one may neglect as before retardation and ignore consequently
terms higher order than $1/c^{2}$. Within this approximation one remains only
with an approximate Hamiltonian being the sum of the $1\over c^{2}$ e.m
Hamiltonian Eq. 8 (as well as some other interactions) for electrons and of
the free Hamiltonian of the photons both in the averaging over equilibrium as
in the time evolution.

Performing the commutations and taking again into account the translation,
rotation and reflection invariance (in the coordinate space) we get after some
algebra

$\displaystyle\tilde{\delta}_{n_{{\vec{k}}}}(\omega)=2\frac{\coth(\frac{\beta\hbar\omega}{2})}{ck\omega}\Re\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\\!\\!\\!\\!dte^{-\imath\omega
t}\\!\\!\\!\int_{0}^{\beta}\\!\\!\\!\\!\\!\\!d\lambda\langle\tilde{j}({\vec{k}},0)j_{T}(0,t+\imath\hbar\lambda)\rangle_{0}^{e}\times$
$\displaystyle\big{[}e^{-\imath(\omega-ck)t-\ ck\lambda}(1+{\cal
N}_{k})+e^{-\imath(\omega+ck)t+ck\lambda)}{\cal N}_{k}\big{]}\enspace,$ (112)

where

${\cal N}_{k}\equiv\frac{1}{e^{\beta kc}-1}$ (113)

is the Bose distribution of photons.

In the above expression again the electronic current-current correlator
appears as in the case of the noise of the transverse electric field, however
they differ essentially due to the presence of the Bose functions under the
integrals. Therefore, it cannot be related to the transverse conductivity.
Nevertheless, Eq. 112 may be the starting point for the direct computation of
the photon number noise spectrum within given solid state models or mesoscopic
systems.

The thermal noise spectra of the longitudinal and transverse electric fields
together with this last discussed case of photon number fluctuations exhaust
the possible thermal noise measurements

##  11 Conclusions.

We have discussed in some details the implementation of electromagnetism in
quantum mechanics. The purpose of this analysis was to avoid the
misinterpretations persistent in the treatment of the magnetic field in solid
state theory. The basic ideas are length well-known, nevertheless there is a
total confusion in the literature about the distinction between the magnetic
field in the matter and the applied classical macroscopic (external) magnetic
field. Another misunderstanding even at the textbook level is about gauge
transformations in the Hamiltonian. We stress also the essential role of the
classical, macroscopic external fields in the description of any
electromagnetic experiment.

In order to bring clarity in this important matter one has to drop the old-
fashioned presentation of the quantum mechanics of electromagnetically
interacting charged particles based on the classical theory of point-like
charges. One must choose the formalism of the quantum field theory in a non-
relativistic frame. This is mandatory, since in condensed matter one has to do
with ions (or nuclei) that are not elementary particles, therefore one cannot
resort to the relativistic QED. The non-relativistic QED was length used in
quantum optics, but not yet in many-body theories of condensed matter. An
intermediary role might play the $1/c^{2}$ approximation of this non-
relativistic QED that separates the motion of the electrons from that of the
photons. The so obtained Coulomb gauge Hamiltonian includes the microscopic
variant of the Biot-Savart law and is in concordance with the 100 year old
proposal of Darwin [1]. We show, that a proper discussion of superconductivity
and of the electromagnetic noise spectra may not ignore this ingredient.

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