We prove that the flee Euclidean electromagnetic field, as constructed by L. Gross .in [2], has the Markoffproperty.
In [2] L. Gross constructed the Euclidean formalism for the free electromagnetic field and produced some arguments implying that the Markoff property would not be true for this model. The result of this paper shows this conjecture not to be true. We shall briefly recall the model introduced by Gross.
We shall use the following notation: K = K a for any space K of ( complex) distributions on the open set f2 C IR a. fdenotes the Fourier transform o f f E 6P'(IR a) or f E ,9"(IRa). We work in d-dimensional space-time (d = s + 1, s = 1, 2, 3, ...) and denote by (a, b) = Z]=o~bj the Euclidean scalar product, [al = ~ a). For any real o let us define the following space of distributions:
Re(IRa) = / r E 6a'(1Ra)I f E Lloc(lR a) and ']fl' o -l/iRalk 12. 'J~(k,l 2 dkI 1/2 0 these spaces are slightly larger than the corresponding Sobolev spaces; ifo < 0, they are smaller). There is a natural anti-duality between R~ a) and R-~ a) defined by:
This is extended to R a β’ R -~ in the usual way, by summing over the d components. It is easily proved (using Fatou lemma) that Ra(IR d) is a Hilbert space if o < all2 (if o I> d/2 then it is not complete) with the norm [l' II a and, if I el 7> all2, then the above antiduality determines a canonical isomorphism between the antidual Ra(iRa) ' and R-a(IRa); in this case we shall identify them.
The Euclidean one-particle Hilbert space is defined as follows (we suppress the index e used by Gross):