Question

When one studies representations of (bosonic) Lie groups in physics, whether dealing with spacetime symmetries or gauge symmetries, it is often left implicit whether the representations are over real or complex vector spaces, without acknowledging of course that the representations of an algebra or group can be different over different base fields. A simple example is that the adjoint representation of the Lorentz group is irreducible as a representation over the reals, but reducible when take as a representation over complex vector spaces.

When it comes to spinors, often people wish to treat components of spinors as Grassmann variables - this being the 'classical limit' of the fermion anti-commutation relations. My question is whether or not there has been a rigorous treatment of the representations of semi-simple (bosonic) Lie groups over Grassmann-valued vector spaces.

EDIT:

I suppose the direct analogy I anticipated isn't there. When we allow complex fields, we also allow complex changes of basis. This changes the reducibility of representations, because real Lie algebra elements now can have imaginary components even though the boost/rotation parameter remains real. For Grassmann numbers, it appears that we dont extend the reals as in the complex case, but supercede them. So we dont gain access to a new change of basis like before, where the algebra elements would have Grassmann-valued components while keeping the boost parameters real. Unless I am mistaken and people do work with spinors with real parts and Grassmann parts.

Answer 1

The question explores the formal treatment of representations of semi-simple Lie groups over Grassmann-valued vector spaces in physics, the key to understanding fermionic behavior in quantum mechanics and field theory, as well as how these representations differ from those over real or complex numbers.

Step-by-step explanation:

The question pertains to the representations of semi-simple Lie groups over Grassmann-valued vector spaces in the context of theoretical physics, specifically within the domain of quantum mechanics and field theory. In standard formulations of quantum mechanics, particles are classified by their intrinsic spin: fermions have half-integer spins and obey the Pauli exclusion principle, while bosons have integer spins and do not adhere to this exclusion. Regarding the representations of Lie groups, it is acknowledged that Grassmann numbers, used to describe fermionic degrees of freedom at a classical level, do not extend the real numbers like complex numbers do but rather supersede them, creating a different algebraic structure that might affect the behavior of group representations and the application of changes of basis within this context.

The question also touches upon the nature of spacetime symmetries and gauge symmetries, represented by Lie groups, and the different characteristics observed when transformations such as Lorentz transformations are considered over real or complex numbers, emphasizing the impact on reducibility of representations. The reducibility aspect is significant since it determines how such symmetries can be decomposed into simpler building blocks or 'irreducible components', each carrying the fundamental properties pertinent to the physical system under consideration.