The magnetic Weyl calculus

Abstract

In the presence of a variable magnetic field, the Weyl pseudodifferential calculus must be modified. The usual modification, based on "the minimal coupling principle" at the level of the classical symbols, does not lead to gauge invariant formulae if the magnetic field is not constant. We present a gauge covariant quantization, relying on the magnetic canonical commutation relations. The underlying symbolic calculus is a deformation, defined in terms of the magnetic flux through triangles, of the classical Moyal product. from the Schwartz space S(R N ) to its dual. This is based on a study of the distribution kernel of this operator. These and some other fundamental facts are also considered in Section 3.Beyond these operators lies a symbolic calculus which is manifestly gauge invariant, being defined only in terms of the magnetic field. The composition is a magnetic correction of the Moyal product, while the involution is just the usual complex conjugation of functions. In Section 4 we study this symbolic calculus. Once again an important problem is to extend the formulae when obvious integrability conditions are not satisfied. This is a more complicated task for the product than it was for the quantization itself, especially if one aims at obtaining * -algebras. We postpone the application of the machinery of oscillatory integrals and classical symbol function spaces to a future article. For our present purposes the strategy of extension by duality methods (see [12], [13] for the non-magnetic case) is more fruitful. It will lead to a large, interesting * -algebra of distributions which will be called the magnetic Moyal algebra.Of course, for certain problems, a well-justified norm on (restricted) * -algebras of symbols could be very useful. It happens that this is easier to achieve after performing a partial Fourier transform. Surprisingly, one naturally encounters certain C * -algebras which were studied in pure mathematics, with little connection with physics. These are special types of twisted crossed products, associated to twisted actions of R N on suitable abelian C * -algebras of position observables. They were already related to quantum magnetic fields in [23]; see also [11], [5] and [6] for related works. In a future publication we shall extend their study and outline the connection with our pseudodifferential calculus.Concerning the difference between the expressions (0.2) (or (0.3)) and (0.4) some comments are necessary. In Subsection 3.4 we shall outline some situations when they give the same result. By admitting the convenient assumption that the components of A are smooth functions with tempered growth, both (0.2) and (0.4) can be extended to any tempered distribution f . Then we shall have Op A (f ) = Op A (f ) for all f if and only if A is linear (this is one of the most important cases appearing usually in the literature). Remark that this condition corresponds to a constant magnetic field, but it is not gauge invariant: for some other A ′ with dA ′ = B = const, Op A ′ and Op A ′ will be diffe...

On the Lipschitz Continuity of Spectral Bands of Harper-Like and Magnetic Schrödinger Operators

“…These phases appear very naturally in the continuous case, see [7,8,19,22,23,24,25,26], where it is shown that if a(x) is the transverse gauge generated by a globally bounded magnetic field |b(x)| ≤ 1, then φ(x, x ′ ) can be chosen to be the path integral of a(x) on the segment linking x ′ with x. This is the same as the magnetic flux of b through the triangle generated by x, x ′ and the origin.…”

Two-parameter asymptotics in magnetic Weyl calculus

“…It turns out that the integral formula for the product found in [MP04,IMP07] is not amenable to the derivation of an asymptotic expansion in ǫ and λ. Although an asymptotic expansion for ǫ = 1 = λ has been derived in [IMP07], calculating each term has proven to be very tedious and it is not obvious how to collect terms of the same power in ǫ and λ.…”

“…However, we can show the equivalence to the product formula obtained by two of the authors in [MP04] by writing out the symplectic Fourier transforms, RHS of (2.6) = 1…”

Cotangent bundle quantization: entangling of metric and magnetic field

“…In the next section we introduce a concrete example of a normal subalgebra F ⊂ L and its extension F * suitable for quantizing the phase space T * M. In the case M = R n similar extensions were used in [15,[61][62][63][64].…”

“…The magnetic algebra generated by relations (1.2) is an interesting and useful object for physical and mathematical applications, [6][7][8][9][10][11][12][13][14][15]. In particular, the case of quadratic magnetic field F represents an example of quadratic quantum algebra (1.2) which corresponds to the symplectic space R 2n of constant non-zero curvature [5].…”