Unique basis of relativistic field equations for arbitrary spin?

[tom.stoer]

Looking at Lagrangians and field equations for different spin all the derivations seem to lack a common basis; they appear to lack any deep relation. Is there a unique way to understand the different forms like Klein-Gordon, Dirac, Maxwell (Yang-Mills), etc. from a common basis which is valid for arbitrary spin?

 

[fzero]

Looking at Lagrangians and field equations for different spin all the derivations seem to lack a common basis; they appear to lack any deep relation. Is there a unique way to understand the different forms like Klein-Gordon, Dirac, Maxwell (Yang-Mills), etc. from a common basis which is valid for arbitrary spin?

The modern presentation, which I'm sure you're aware of is to find the field that provides an irreducible representation of the Poincare group. Then one writes down all Poincare invariant terms and forms a Lagrangian. The coefficients are arbitrary, except one usually chooses a conventional normalization for the kinetic term. If you want a renormalizable theory in 4d, then you restrict to dimension 4 operators. You can also impose further global symmetries if you wish. Quantization of the spin 1 field will require gauge invariance.

These rules lead to field equations which look very different, but that is a consequence of Poincare covariance. The deep relation between them, at least formally, is supersymmetry.

 

[tom.stoer]

The modern presentation, which I'm sure you're aware of is to find the field that provides an irreducible representation of the Poincare group. Then one writes down all Poincare invariant terms ...

You have to do that for each spin 0, 1/2, 1, 3/2, ... seperately and you don't get any relations between them. Thanks for the SUSY/SUGRA idea, perhaps this is the only way out.

 

[fzero]

You have to do that for each spin 0, 1/2, 1, 3/2, ... seperately and you don't get any relations between them. Thanks for the SUSY/SUGRA idea, perhaps this is the only way out.

Yes, you are asking for a symmetry that relates superselection sectors of the Lorentz group. These are constrained by the Coleman-Mandula theorem to be supersymmetries. I suppose that I have encountered other options in theories where the assumptions of the CM theorem are relaxed, but I haven't studied them.

 

[garrett]

Hi Tom,

You're after a unified description of scalar, fermion and gauge fields… very ambitious. But don't forget the gravitational spin connection and frame.

Let $$A$$ be a 1-form gauge field valued in a Lie algebra, say spin(10) if you like GUTs, and $$\omega$$ be the gravitational spin connection 1-form valued in spin(1,3), and $$e$$ be the gravitational frame 1-form valued in the 4 vector representation space of spin(1,3), and let $$\phi$$ be a scalar Higgs field valued in, say, the 10 vector representation space of spin(10). Then, avoiding Coleman-Mandula's assumptions by allowing e to be arbitrary, possibly zero, we can construct a unified connection valued in spin(11,3):
$$H = {\scriptsize \frac{1}{2}} \omega + \frac{1}{4} e \phi + A$$
and compute its curvature 2-form as
$$F = d H + \frac{1}{2} [H,H] = \frac{1}{2}(R - \frac{1}{8}ee\phi\phi) + \frac{1}{4} (T \phi - e D \phi) + F_A$$
in which $$R$$ is the Riemann curvature 2-form, $$T$$ is torsion, $$D \phi$$ is the gauge covariant 1-form derivative of the Higgs, and $$F_A$$ is the gauge 2-form curvature -- all the pieces we need for building a nice action as a perturbed $$BF$$ theory. To include a generation of fermions, let $$\Psi$$ be an anti-commuting (Grassmann) field valued in the positive real 64 spin representation space of spin(11,3), and consider the "superconnection":
$$A_S = H + \Psi$$
The "supercurvature" of this,
$$F_S = d A_S + A_S A_S = F + D \Psi + \Psi \Psi$$
includes the covariant Dirac derivative of the fermions in curved spacetime, including a nice interaction with the Higgs,
$$D \Psi = (d + \frac{1}{2} \omega + \frac{1}{4} e \phi + A) \Psi$$
We can then build actions, including Dirac, as a perturbed $$B_S F_S$$ theory.

Once you see how all this works, the kicker is that this entire algebraic structure, including spin(11,3) + 64, fits inside the E8 Lie algebra.

 

[mitchell porter]

Tom, are you interested only in free fields and only in four dimensions?

 

[tom.stoer]

Tom, are you interested only in free fields and only in four dimensions?

No, of course not.

My observation was that the construction of Lagrangians for different spin is - to a large extend - arbitrary. Certain constraints arise during quantization (due to renormalization, spin < 5/2, ...) but nevertheless there is a huge freedom in picking certain terms. There should be some underlying construction principle which "produces" or "predicts" the various terms.

I know that SUGRA seems to be a rather predictive framework regarding allowed structures in different dimensions but my idea was that there could be some other principle besides SUGRA (and perhaps string theory), but I see that your ideas again point into this direction.

 

[negru]

Well, there are a few models for higher spin theories, some with susy, some without, coming from the same lagrangian. Usually you start with a "classical spinning particle" model, and when you quantize it it's spin will depend on some constraint on the Pauli-Lubanski vector..

For example, the work of Kuzenko (http://arxiv.org/abs/hep-th/9403196, http://arxiv.org/abs/hep-th/9512115), the classic stuff by Fronsdal, Vasiliev (usually for free theories though)...but I'm not sure if this is what you're looking for.

 

[mitchell porter]

Like negru, I don't know how broadly you want to cast your net. The opposite extreme from "only free fields, only four dimensions" would be to look at all interacting field theories in all dimensions. Or, you can stipulate a property, like gauge symmetry or conformal symmetry or integrability, which defines a nontrivially restricted class of field theories, and then you can study that class. But I don't know what sort of property or what sort of restriction you're looking for...