### Abstract

The indefinite metric state space S_{ M } of the covariant form of the quantized Maxwell field *M* contains, as is known, a family of continuously many, isomorphic, isometric pre‐Hilbert spaces L^{ q }, called Lorentz spaces, each of which corresponds to one square‐integrable, prescribed, classical, spatial distribution *q*(x) of the total charge *Q* = 0. The quotient spacesL^{ q }/N^{ q } modulo the subspaceN^{ q }⊆L^{ q } of all elements with norm 0 are indeed Hilbert spaces in S_{ M } and it appears that any QED which has to do with S_{ M } has to be formulated not on S_{ M } as a whole but on the family of these Lorentz spaces. To support this assumption we embed any L^{ q }/N^{ q } in L^{ q } in an isomorphic‐isometric way and thus get Hilbert spacesl^{ q } in S_{ M }, but not in a unique way; we will show that the different possibilities of this embedding correspond exactly with the different gauges. The main results about any embedding space l^{ q } are, however, that it is a *m* *a* *x* *i* *m* *a* *l*Hilbert space in S_{ M } (under a premise referring to the expectation values of charge distribution), and that any Hilbert space of S_{ M } which is ’’physically important’’ in some sense is necessarily one of these l^{ q }. In this way the family {l^{ q }‖*q*∈Q} of Lorentz spaces (defined by the index set Q) has some outstanding properties so that the l^{ q } are now characterized by these qualities (and no longer in the heuristical way via generalization of the classical Lorentz condition). Starting now with the prominent role of the l^{ q } we get not only a deeper understanding of the Lorentz condition of classical electrodynamics—the properties of the l^{ q } lead us automatically to the definition of a positive‐definite state space of QED with the use only of these l^{ q }. (Our considerations refer not to full but to some restricted QED; the restriction is primarily given by the above‐mentioned index set Q so that extensions to full QED seem possible.) We show that our definition of a state space is consistent with time evolution, given by the Hamiltonian *H*, and that the QED on the basis of this state space is a constraint‐free theory because the otherwise‐necessary selection rules of Lorentz condition and charge conservation are now superfluous (and not present in a hidden form either). Furthermore, the properties of Lorentz spaces lead us automatically to a new concept of observables, all of which commute from the beginning with the operator of charge distribution. As a special observable we discuss the number operator *N*(k) of photons by showing that this, in general, cannot be of the form *a* ^{+} _{μ} *a* ^{μ}. A modified form of *N*(k) and with it a reformulation of the Hamiltonian *H* of QED is given. All these considerations go back to the properties of the Lorentz spaces and thus, basically, to the canonical quantization of the Maxwell field.

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