# Stability of Relativistic Quantum Electrodynamics in the Coulomb Gauge

###### Abstract

We show that relativistic quantum electrodynamics in the Coulomb gauge satisfies the following bound, which establishes stability: let denote the Hamiltonian of on the three-dimensional torus of volume and with ultraviolet cutoff . Then there exists a constant (the vacuum energy renormalization) such that the renormalized Hamiltonian is positive:

## 1 Introduction

The proof of stability of non-relativistic matter interacting with a classical electromagnetic field is one of the crown jewels of mathematical physics, both from the point of view of physics and mathematics (see [13, Chapters 1-7 & 9] for a comprehensive exposition and references). It accounts for a wide, enormously rich class of phenomena in quantum mechanics, which are of crucial importance to the macroscopic world and even to everyday life.

There are, however, various phenomena which require the quantization of the electromagnetic field, such as spontaneous emission [14] and the black-body radiation, with its astoundingly perfect fit to the spectrum of the cosmic radiation background [18]. In addition, there is the well-known conceptual necessity to quantize the electromagnetic field [3], an argument which does not extend to the gravitational field (as Dyson recently remarked [5]).

Quantization of the electromagnetic field is, however, a well-known source of trouble. Its coupling to non-relativistic matter has been studied extensively, see [13, Chapter 11, and references given there] and Spohn’s treatise [15], Part II, Chapters 19 and 20, and also the references he cites. One particular important step was taken by Lieb and Loss [10] (see also [15, pp. 314–315]), who established an upper bound to the ground state energy. As the latter disagrees with the result suggested by perturbation theory, the bound by Lieb and Loss implies that perturbation theory can not converge: it simply would yield a wrong picture of the electron cloud (see also [9] for an illuminating discussion).

For non-relativistic matter interacting with the radiation field,
the term in the Hamiltonian, where stands for
the momentum operator of a particle, and for the quantized electromagnetic vector
potential field with ultraviolet cutoff , *seems to indicate* a lower bound to the Hamiltonian only of
type , with proportional to the number of static nuclei. Fröhlich [6] has
remarked that, while such a bound proves stability of matter if an ultraviolet cutoff is imposed on the theory,
the linear dependence on is disastrous, physically speaking. He raised the question whether such
a catastrophe does indeed prevail, relating it to Landau’s conjectures (the so-called Landau pole, see [8]).
He also remarked, at the time (and in this respect the situation
has not changed since) that *it was not known* (provided a mass renormalization and a chemical potential
renormalization are chosen appropriately) whether a lower
bound on can be found which is *uniform* in .

Similar problems are expected in the case of relativistic quantum electrodynamics in the Coulomb gauge, where the term plays a role similar to the above mentioned term , where now denotes a (regularized) electron-positron current, but where, in addition, a charge renormalization is expected from perturbation theory. In this paper we show that for the Hamiltonian of relativistic quantum electrodynamics in the Coulomb gauge a suitable vacuum energy renormalization can be found such that the renormalized Hamiltonian (with no mass and no charge renormalization) is positive. In doing so, we provide a more complete picture of the electron cloud. It affects both the effective interaction between the electrons, and their kinetic energy, yielding a picture of “dressed” electrons and positrons.

## 2 Qed on the Three-Torus

We will study quantum electrodynamics in the Coulomb gauge, with a cut-off Hamiltonian

(1) |

acting on a Hilbert space that is the tensor product of an antisymmetric Fock space and a symmetric Fock space . The one-particle space for both Fock spaces is , with

Here is a *finite* set, symmetric
under inversion about each coordinate plane containing the origin, and therefore invariant
under inversion through
the origin. The number of sites in will be denoted by .
Since is finite, there are no antisymmetric particle functions for .
Hence, Pauli’s principle ensures that is finite-dimensional;
however, this argument does not hold for bosons, and is, in fact,
infinite-dimensional.

The first two operators on the r.h.s. in (1) denote the free massiv fermion Hamiltonian

(2) |

and the free massless boson Hamiltonian

(3) |

respectively. The constant appearing in (2) is the electron mass. The photon (resp. electron and positron) annihilation and creation operators (resp. , , , ) are normalized so that

and

where is the Kronecker delta. All other commutators (or anti-commutators) are zero. The operators and depend on (but not on ). In the sequel, however, we frequently deal with operators that depend on both and , and so it will be convenient to set .

The *photon-fermion interaction*, with periodic boundary conditions, is

(4) |

The dot denotes the scalar product of three-vectors in Euclidean space. The
*electric current*
appearing in (4) is a three-vector-valued operator on , whose components can be
expressed, using the Pauli matrices, in terms of the electron-positron
fields :

(5) |

is a -matrix for each .
The symbol . The star on an operator denotes adjoint and the dagger above in the definition of the
current does not change the four vectors, in conformance with the above reference to the scalar product in , see below.
The *electron-positron field* itself is given by denotes the
scalar product in

(6) |

(7) |

with four-vectors

The Fermi field is a *bounded* operator for each ,
but this statement does not hold for
the vector potential appearing in (4). The latter is given by

(8) |

The
*polarisation vectors* can be chosen (see [11]) such that for , ,

(9) |

and , as well as . The question of the choice of polarization vectors in qed is a nontrivial matter, see also [12], and the forthcoming (38). Note that the three vectors form an oriented orthonormal basis in . Hence, by (8),

which characterises *Coulomb gauge*.
The final term in (1) represents the *Coulomb interaction*

(10) |

We note that, with the above notation, denotes the instantaneous Coulomb interaction *without* normal
ordering, *i.e.*, without the Wick dots. Here denotes a regularized Coulomb potential on the torus:

(11) |

In the limit tends to the Coulomb potential in the distributional sense. Note that due to the exclusion of in (8) .

It was proven in [7] that there exists a dense set of vectors in on which is essentially self-adjoint. The closure of has a purely discrete spectrum with finite multiplicity, it is bounded from below, and the eigenfunctions of lie in .

## 3 A finite-dimensional Grassmann algebra

In the sequel, will denote either an electron or a positron creation operator. To every vector , written in the form

(12) |

we will now associate an element

of the Grassmann algebra generated by anti-commuting symbols and :

(13) |

with and . The algebra is of dimension ; see [1, p. 49]. The set of ’s corresponding to the state vectors is denoted by .

To further simplify the notation, we enumerate the momenta in , and the set

Next, one introduces symbols , subject to the commutation relations

(14) |

and defines single integrals

(15) |

Multiple integrals are understood as iterated integrals. Thus, (14) and (15) define the integral

for all monomials and one can than extend the integrals to arbitrary elements by linearity.

The Grassmann algebra has an involution [1, p. 66] and, corresponding to the inner product in , there is an associated inner product in

(16) |

where and .

###### Lemma 3.1.

We next discuss how a bounded operator acting on is represented
by an element ,
which naturally acts on . The connection is particularly simple, if is given in its *normal form*

(17) |

as this allows us to represent by (see [1, p. 26])

(18) | ||||

In fact, if , then corresponds to the element (using a suggestive notation)

see [1, Equ. (3.68), p. 84].

###### Remark 3.1.

## 4 The first unitary transformation

In this section, we apply a unitary transformation to the Hamiltonian

(20) |

where both

and

act trivially on the bosonic Fock space .

Inspecting the parts of , which involve bosonic creation and annihilation
operators, *i.e.*,

one may try to “complete the square” by adding and subtracting a term of the form

which is quadratic in . In fact, there is a unitary operator, namely

which accomplishes this task:

###### Proposition 4.1.

As a quadratic form on ,

(21) |

where the positive first term on the r.h.s. is build up from *dressed* bosonic creation and annihilation operators

and

Moreover,

(22) |

###### Remark 4.1.

Note that the action of and on is no longer trivial, as mixes the components in the tensor product .

###### Proof.

The are Grassmann variables, which commute with all the other Grassmann variables. Thus, (21) is a consequence of

The final statement follows from the fact that on , the Grassmann variables commute. Hence, . ∎

In the original representation on the current-current interaction is represented (using the correspondence (17)–(18)) by the normal product of

(23) |

Recalling (5), we conclude that the difference

(24) |

consists of a sum of four terms

(25) |

obtained by replacing terms of the form and occurring in by the identity operator : let denote the matrix

then

with coefficients

(26) |

Note that , , .

An interesting aspect of the explicit form of given in (23) is that is shows a certain similarity to the Coulomb interaction introduced in (10). We can explore this fact, by establishing an inequality which generalizes the positivity bound for the Coulomb energy:

###### Proposition 4.2.

###### Proof.

We may write in the form

By a unitary transformations on can convert each of the four summands in to a direct product of diagonal matrices . The eigenvalues of each are zero and two, each with multiplicity two, as may be verified by straightforward diagonalisation, independent of , because they depend only on the Euclidean norms and , which are both equal to one. The inequality (28) then follows. ∎

###### Corollary 4.3.

.

###### Proof.

We will also need a bound on general Fermi operators.
As before, the Fermi creation operators and annihilation operators
will satisfy
canonical anti-commutation relations, *i.e.*,
and the remaining anti-commutators are all equal to zero.

###### Lemma 4.4 (Bogoliubov & Bogoliubov [2]).

Let be a self-adjoint fermionic Hamiltonian of the general form

(29) |

where the matrix is hermitian
and the matrix is anti-symmetric, *i.e.*,

Then there exists a constant such that .

###### Proof.

It has been shown in [2] that a Hamiltonian of the form (29) can always be written as

(30) |

with new Fermi creation and annihilation operators

(31) |

and the new Fermi operators satisfying canonical anti-commutation relations. The matrices and appearing in (31) satisfy

(32) |

and is given by and denotes the diagonal matrix of the . ∎

###### Theorem 4.5.

There exists a real number (the vacuum energy renormalization) such that the first renormalized Hamiltonian

where , the original Hamiltonian describing qed in the Coulomb gauge, introduced in (1), satisfies

(33) |

as a quadratic form on .