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On Quantum Mechanics with a Magnetic Field on ℝn and on a Torus \(\mathbb{T}^{n}\), and Their Relation

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Abstract

We show in elementary terms the equivalence in a general gauge of a U(1)-gauge theory of a scalar charged particle on a torus \(\mathbb{T}^{n}=\mathbb{R}^{n}/\varLambda\) to the analogous theory on ℝn constrained by quasiperiodicity under translations in the lattice Λ. The latter theory provides a global description of the former: the quasiperiodic wavefunctions ψ defined on ℝn play the role of sections of the associated hermitean line bundle E on \(\mathbb{T}^{n}\), since also E admits a global description as a quotient. The components of the covariant derivatives corresponding to a constant (necessarily integral) magnetic field B=dA generate a Lie algebra g Q and together with the periodic functions the algebra of observables \(\mathcal {O}_{Q}\). The non-abelian part of g Q is a Heisenberg Lie algebra with the electric charge operator Q as the central generator; the corresponding Lie group G Q acts on the Hilbert space as the translation group up to phase factors. Also the space of sections of E is mapped into itself by g∈G Q . We identify the socalled magnetic translation group as a subgroup of the observables’ group Y Q . We determine the unitary irreducible representations of \(\mathcal{O}_{Q},Y_{Q}\) corresponding to integer charges and for each of them an associated orthonormal basis explicitly in configuration space. We also clarify how in the n=2m case a holomorphic structure and Theta functions arise on the associated complex torus.

These results apply equally well to the physics of charged scalar particles on ℝn and on \(\mathbb{T}^{n}\) in the presence of periodic magnetic field B and scalar potential. They are also necessary preliminary steps for the application to these theories of the deformation procedure induced by Drinfel’d twists.

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Notes

  1. As the holomorphic structure w.r.t. the complex variables z j=x j+ix m+j is not invariant under x↦gx for generic g∈GL(n), the choice Λ=2πℤn would be a loss of generality if n=2m and we were concerned with holomorphic line bundles on the complex m-torus \(\mathbb{T}^{n}=\mathbb{C}^{m}/\varLambda\). See also the end of Sect. 3.

  2. Actually it is not necessary to assume from the start that V∈U(1); assuming just that it is nonvanishing complex, V∈U(1) will follow from the reality of A a , see below.

  4. As \(V^{-1}(l +l',x)V(l,x+2\pi l') V(l',x)= e^{-iq4\pi^{2}l^{t}\beta^{A}l'}\), (3) amounts to q2πl t β A l′∈ℤ for all q∈ℤ and l,l′∈ℤn, i.e. to (9).

  5. In fact,

    The second equality holds because dB=0, the third by Stokes theorem, the fourth by the periodicity of A′,B, which makes the border integral vanish.

  6. For instance, imposing the conditions (2)–(3) for all q∈ℤ and only for l,l′ such that l 1=kh 1, \(l_{1}'=kh_{1}'\) (with some \(h_{1},h_{1}'\in\mathbb{Z}\)) would lead to kν 1b ,kν b1∈ℤ i.e. ν 1b ,ν b1∈ℚ, and again to ν ab ∈ℤ for a,b>1. The corresponding ψ’s could be interpreted as k-component wavefunctions, i.e. sections of a \(\b{C}^{k}\)-vector bundle.

  7. Namely, for all \(c\in\mathcal{O}_{Q}\), \(\psi\in \mathcal{X}^{V}\), \(f\in\mathcal{X} \), g∈g Q g▷(cψf)=(g▷c)ψf+c(g▷ψ)f+cψ(g▷f).

  8. $$ e^Re^S=e^{R+S}e^{-\frac{1}{2}[R,S]},\quad\mbox{if } [R,S] \mbox{ commutes with } R,S. $$
    (17)

  9. Equation (46) is verified by direct inspection for (n,k)=(0,0), using the relation \(m_{2}f(x^{1},x^{2}) = e^{ix^{1}+i\frac{\pi}{\tilde{\nu}}\tilde{\alpha}_{2}} f (x^{1},x^{2} + \frac{\pi}{\tilde{\nu}} )\) [valid for all f∈C ∞(ℝ)] and (45).

  10. The whole commutant (centralizer) \(\tilde{M}\) of G within Y is the subgroup

    $$\tilde{M}=M M',\qquad M' := \Biggl\{ \exp \Biggl[ih^0 + i\sum_{a=2r + 1}^np_a z^a \Biggr] \mid \bigl(h^0 , z^{2r + 1} ,\ldots, z^n\bigr)\in\mathbb{R}^{n - 2r + 1} \Biggr\} $$

    Equivalently, [m,U g]=0, i.e. [m,p a ]=0, for all \(m\in\tilde{M}\).

  11. The points x∈W j , x′∈W i such that u=P j x=P i x′ are related by x′=x+2πl, with some l∈ℤn. One has just to replace the arguments l,x of V in (2) resp. by \(P_{i}^{-1}(u)-P_{j}^{-1}(u)\), \(P_{j}^{-1}(u)\).

  12. The points x∈U k , x′∈W j , x″∈W i such that [x]=[x′]=u=P i x″=P j x′=P k x are related by x′=x+2πl′, x″=x′+2πl with some l,l′∈ℤn. One has to replace x,x+2πl′,l,l+l′ in (3) resp. by \(P_{k}^{-1}(u),P_{j}^{-1}(u), [P_{i}^{-1}(u)-P_{j}^{-1}(u) ]/2\pi, [P_{i}^{-1}(u)-P_{k}^{-1}(u) ]/2\pi\), and use the above definition of t ij .

  13. It must be \(\tilde{W}_{i}=W_{i}+ 2\pi l_{i}\) for some l i ∈ℤn, whence \(\tilde{\boldsymbol{\psi }}_{i}(u)=\psi [\tilde{P}_{i}^{-1}(u) ]=\psi[P_{i}^{-1}(u)+2\pi l_{i} ]=\) \(V [l_{i},P_{i}^{-1}(u) ]\psi[P_{i}^{-1}(u) ]= \tilde{U}_{i}(u)\boldsymbol{\psi}_{i}(u)\), where \(\tilde{U}_{i}(u):=V [l_{i},P_{i}^{-1}(u) ]\).

  14. As P(x+z)=T [z] u∈X j , then \(x+z=P_{j}^{-1} (T_{[z]} u )\), whereas \(x=P_{i}^{-1}(u)\in X_{i}\); replacing these formulae in (18) we obtain the second equality in (68). As a consistency check, it is straightforward to verify that the conditions \([g_{\tilde{z}}\psi]_{i}=t_{ij} [g_{\tilde{z}}\psi]_{j}\) are satisfied.

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Acknowledgements

It is a pleasure to thank D. Franco, J. Gracia-Bondía, F. Lizzi, R. Marotta, F. Pezzella, R. Troise, P. Vitale for useful discussions. We acknowledge support by the “Progetto FARO: Algebre di Hopf, differenziali e di vertice in geometria, topologia e teorie di campo classiche e quantistiche” of the Universita’ di Napoli Federico II.

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  1. Dip. di Matematica e Applicazioni, Università “Federico II”, V. Claudio 21, 80125, Napoli, Italy

    Gaetano Fiore

  2. I.N.F.N., Sez. di Napoli, Complesso MSA, V. Cintia, 80126, Napoli, Italy

    Gaetano Fiore

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Fiore, G. On Quantum Mechanics with a Magnetic Field on ℝn and on a Torus \(\mathbb{T}^{n}\), and Their Relation. Int J Theor Phys 52, 877–896 (2013). https://doi.org/10.1007/s10773-012-1396-z

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