The uncertainty of fluxes

In the ordinary quantum Maxwell theory of a free electromagnetic field, formulated on a curved 3-manifold, we observe that magnetic and electric fluxes cannot be simultaneously measured. This uncertainty principle reflects torsion: fluxes modulo torsion can be simultaneously measured. We also develop the Hamilton theory of self-dual fields, noting that they are quantized by Pontrjagin self-dual cohomology theories and that the quantum Hilbert space is Z/2Z-graded, so typically contains both bosonic and fermionic states. Significantly, these ideas apply to the Ramond-Ramond field in string theory, showing that its K-theory class cannot be measured. Fluxes in the classical theory of electromagnetism and its generalizations are real-valued and Poisson-commute. Our main result is a Heisenberg uncertainty principle in the quantum theory: magnetic and electric fluxes cannot be measured simultaneously. This observation applies to any abelian gauge field, including the standard Maxwell field theory in four spacetime dimensions as well as the B-field and Ramond-Ramond fields in string theories. It is the torsion part of the fluxes which experience uncertainty - the nontrivial commutator of torsion fluxes is computed by the link pairing on the cohomology of space, and there are always nontrivial commutators if torsion is present. We remark that torsion fluxes arise from Dirac charge/flux quantization. This Heisenberg uncertainty relation goes against the conventional wisdom that the quantum Hilbert space is simultaneously graded by the abelian group of magnetic and electric flux; in fact, it is only graded by the free abelian group of fluxes modulo torsion. The most interesting example is the Ramond-Ramond field in 10-dimensional superstring theory. Here the Dirac quantization law is expressed in terms of topological K-theory, and conventional wisdom holds that the quantum Hilbert space is graded by the integer K-theory of space. The main result proves this wrong: the grading is only by K-theory modulo torsion. Notice that there are still operators reflecting the quantization by the full K-theory group; the assertion is that these operators do not all commute among themselves if there is torsion. Our exposition in Section1 begins with the classical Maxwell equations. We work on a compact(1) 3-dimensional smooth manifold Y. We first define the classical fluxes and show that they Poisson commute. The Hamiltonian formulation of Maxwell's equations has Poisson brackets which are not invertible, so do not derive from a symplectic structure, if the second cohomology of Y is nontrivial: the symplectic leaves of the Poisson structure are parametrized by the fluxes. The most natural quantization of this system is as a family of Hilbert spaces parametrized by the real vector space of fluxes. Dirac charge quantization is implemented in Maxwell theory by writing the electromagnetic field as the curvature of a T-connection, where T = U(1) is the circle group. There is now an action principle and the space of classical solutions is a symplectic manifold, the tangent bundle to the space C(Y) of equivalence classes of T-connections on Y. Its quantization is a single Hilbert space, defined as the irreducible representation of the Heisenberg group built from the product of C(Y) and its Pontrjagin dual. (The salient features of Heisenberg groups and their representations are reviewed in Appendix A.) Magnetic and electric fluxes are refined to take values in the abelian group H-2(Y; Z). The Heisenberg uncertainty relation, stated in Theorem 1.6, follows from the commutation relations in the Heisenberg group. Our second aim in this paper, carried out in Section2, is to establish an appropriate Hamiltonian quantization of generalized self-dual fields, such as the Ramond-Ramond field in superstring theory. We highlight the main issues with the simplest self-dual field: the left-moving string on a circle, which we simply call the self-dual scalar field. Its quantization, which does not quite follow from the usual general principles, serves as a model for the general case. The flux quantization condition for other gauge fields, both self-dual and non-self-dual, is expressed in terms of a generalized cohomology theory. The fields themselves live in a generalized differential cohomology theory. We briefly summarize the salient points of the differential theory. The data we give in Definition 2.2 is sufficient for the Hamiltonian theory developed here; the full Lagrangian theory requires a more refined starting point. As in the Maxwell theory one can write classical equations (2.10) and a Heisenberg group (Theorem 2.3), which now is Z/2Z-graded. The Hilbert space of the self-dual field is defined (up to noncanonical isomorphism) as a Z/2Z-graded representation of that graded Heisenberg group. The fact that the Hilbert space is Z/2Z-graded, so in general has fermionic states, is one of the novel points in this paper. The noncommutativity of quantum fluxes (2.12) in the presence of torsion is then a straightforward generalization of Theorem 1.6 in the Maxwell theory. Section 2 concludes by showing how some common examples, including the Ramond-Ramond fields, fit into our framework. We call attention to one feature which emerged while investigating self-dual fields in general. For non-self-dual generalized abelian gauge fields any generalized cohomology theory may be used to define the Dirac quantization law. However, for a self-dual field the cohomology theory must itself be Pontrjagin self-dual. See Appendix B for an introduction to generalized cohomology theories and duality. Pontrjagin self-duality is a strong restriction on a cohomology theory. Ordinary cohomology, periodic complex K-theory, and periodic real K-theory(2) are all Pontrjagin self-dual and all occur in physics as quantization laws for self-dual fields. Pontrjagin self-duality is not satisfied by most cohomology theories. For example, if the cohomology of a point in a Pontrjagin self-dual theory contains nonzero elements in positive degrees, then there are nonzero elements in negative degrees as well if the duality is centered about degree zero. In this paper we confine ourselves to the Hamiltonian point of view. We only construct the quantum Hilbert space and the operators which measure magnetic and electric flux up to noncanonical isomorphism. In future work we plan to develop the entire Euclidean quantum field theory of gauge fields, both self-dual and non-self-dual. We emphasize that the Hamiltonian quantization we use for a self-dual field is a special definition; it does not follow from general principles of quantization - see the discussion at the beginning of Section2. Perhaps one should not be surprised that the Hamiltonian theory for self-dual fields requires a separate definition; after all, the same is true for the Lagrangian theory [W]. Our definition is motivated by some preliminary calculations for a full theory of self- dual fields as well as by the special case of the self-dual scalar field in two dimensions. The role of Heisenberg groups and the noncommutativity of fluxes we find in Maxwell theory was anticipated in [GRW]. Our point of view in this paper is unapologetically mathematical. The fields under discussion are free, their quantum theory is mathematically rigorous, whence our mathematical presentation. In particular, we use the representation theory of Heisenberg groups to define the quantum Hilbert space of a free field. Appendix A reviews these ideas in the generality we need. A companion paper [FMS] presents our ideas and fleshes out the examples in a more physical style.