# Legal values of quantum field can take? $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, ..?

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Main issue: What are the legal and possible values of the quantum field can take?

Clarify by examples:

(1) For example, for the spin-0 Klein Gordon field $$\phi$$, we may choose it to be:

• real $$\mathbb{R}$$.
• complex $$\mathbb{C}$$.

(2) For the spin-1/2 fermion field $$\psi$$, we may choose it to be a spinor which needs to be

• Grassman variable

but can also be

• complex $$\mathbb{C}$$. (Dirac or Weyl spinor/fermion)
• We can ask: Can it be in real $$\mathbb{R}$$? (Majorana or Majorana-Weyl spinor/fermion)

(3) For the spin-1/ boson field $$A_\mu$$, we may choose it to be a vector which needs to be

• real $$\mathbb{R}$$ usually for photon field.

• but can it be complex $$\mathbb{C}$$?

(3) How about the spin-3/2 fermion field $$\psi_\mu$$?

• can it be in real $$\mathbb{R}$$, complex $$\mathbb{C}$$, quaternion $$\mathbb{H}$$, ..?
This post imported from StackExchange Physics at 2020-11-24 18:31 (UTC), posted by SE-user annie marie heart
In some theories the field takes values in a Lie group.

This post imported from StackExchange Physics at 2020-11-24 18:31 (UTC), posted by SE-user G. Smith
Not sure why this is put on hold. It's a pretty good question. It boils down to, at least, as far as I can think, studying the stability of a theory resulting from such generalizations. 'Stability' here could mean unitarity, absence of negative energy states, etc. One example I know of a similar generalization is when you make the worldsheet coordinates in string theory non-Archimedean, p-adic valued, giving you p-adic string theory (e.g. sciencedirect.com/science/article/pii/0550321388902076)

This post imported from StackExchange Physics at 2020-11-24 18:31 (UTC), posted by SE-user Avantgarde
@AccidentalFourierTransform I was thinking of the fields in nonlinear sigma models, not gauge fields. These models were not part of my QFT course, so I may completely misunderstand them. I thought the fields in these models have values in Lie groups such as $O(3)$. Is that not the case?

This post imported from StackExchange Physics at 2020-11-24 18:31 (UTC), posted by SE-user G. Smith
@AccidentalFourierTransform The Scholarpedia article “Nonlinear sigma model” (scholarpedia.org/article/Nonlinear_Sigma_model) says that the fields map spacetime into a Riemann target manifold $M$. It then goes on to say, “All the isometries of the target space $M$ form the group $G$ representing the global symmetry of $M$. The most important case is given by the target space that is a Lie group $G$ itself.” So this seems to confirm my statement. Do you agree?

This post imported from StackExchange Physics at 2020-11-24 18:31 (UTC), posted by SE-user G. Smith
I re-ask a different and focused one here: physics.stackexchange.com/q/482858

This post imported from StackExchange Physics at 2020-11-24 18:31 (UTC), posted by SE-user annie marie heart
@all, physics.stackexchange.com/q/482858/42982 physics.stackexchange.com/q/482860/42982

This post imported from StackExchange Physics at 2020-11-24 18:31 (UTC), posted by SE-user annie marie heart
An answer would depend on which model that we're talking about. And there are a lot of models to choose from...

This post imported from StackExchange Physics at 2020-11-24 18:31 (UTC), posted by SE-user Qmechanic
@G.Smith Ah, yes, you're absolutely right. I thought you were referring to gauge fields, thus my comment. But non-linear sigma models do take values in a generic homogeneous manifold, which may indeed be a Lie group. Although strictly speaking, the fields can be thought of as coordinates on the manifold, and so $\mathbb R$-valued. But anyway.

This post imported from StackExchange Physics at 2020-11-24 18:31 (UTC), posted by SE-user AccidentalFourierTransform

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