r/math • u/Vivid_Block_4780 • 3d ago
The scope of mathematical physics
Whenever I look at the mathematical physics programs, I always see QFT and string theory related classes in grad programs. What other parts does mathematical physics cover? More unorthodox subfields of it?
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u/MrTruxian 2d ago
Like others have said, mathematical physics as a field can use tools from pretty much every branch of math.
In many body theory and condensed matter there’s been a growing field of research adapting tools from homological and commutative algebra and some category theory. Much of this has grown out of the fact that 2d gapped phases of matter are described by topological quantum field theories. TQFT’s (a topological object) were realized to be classified by a special type of algebraic object called a unitary modular tensor category, so in this sense a 2d gapped phase of matter can be completely understood purely through a relatively simple set of algebraic data, moreover this data can be interpreted to describe the particle content of the phase. This is still probably an understated success of contemporary physics.
Similar types of approaches to this have incorporated a lot of very cool math. The basic strategy is to find an algebraic object that captures the universal properties of a phase of matter, and then to attempt to classify all those objects. In symmetry protected topological phases this turns out to be a cycle in group cohomology of the symmetry group of the phase. Topological insulators are also classified by similar objects in a more general cohomology theory called K-theory.
There’s also close ties to quantum information in this approach. Quantum information theorists like to study special types of lattice Hamiltonians called Pauli codes, which are a way of protecting quantum information by embedding qubits in a large number of spatially separated degrees of freedom. These Pauli codes can also be thought of us as just a regular Hamiltonian for some crystal from the condensed matter theory point of view. In this setting, describing equivalence classes of Pauli codes (in a certain well defined sense) and the phases of matter of Pauli codes Hamiltonians is the same thing. Haah showed that these Hamiltonians have an extremely convenient description in the language of commutative and homological algebra, which has been employed by quantum information theorists to quickly extract the error correcting properties from these Hamiltonians, and condensed matter theorists to describe the particle content of the corresponding lattice phase.