Worldline formulation of QFT

Earlier I worked on the worldline formalism of quantum field theory for some time so I can answer your question well.

The first quantised worldline technique can be traced back to the appendix of Feynman's foundational paper on the field theory (second quantisation), following which Feynman diagrams and the machinery of field theory became more popular than the first quantised worldline approach -- see Phys. Rev. 80, (1950), 440 (scalar QED) and Phys. Rev. 84 (1951), 108 (spinor QED). It was later picked up by Strassler and substantially developed -- https://arxiv.org/abs/hep-ph/9205205 who rederived the Bern-Kosower rules originally found in infinite tension limit of string theory.

The above cases deal mostly with Abelian scalar and spinor QED, but to return to OP's question, there are extensions to non-Abelian interactions, descriptions of spin 1, 3/2, 2 and higher spin fields as well as differential forms. Fields with spin distinct from 0 generally enjoy a worldline supersymmetry (and further symmetries) that have proven essential in their quantisation.

I found a recent report at https://arxiv.org/abs/1912.10004 and recommend the classical review at Phys. Rept. 355 (2001) 73–234, [https://arxiv.org/abs/hep-th/0101036 ]. The recent report seems to contain plenty of information about higher spin and gravity. See also the various papers such as

• Nucl. Phys. B234, 2, (1984), 269
• arXiv 9504097
• arXiv 9312147
• doi 10.1063/1.1665567
• https://arxiv.org/abs/1905.00945
• arXiv 9801105

for various resources focussed on gravity, or

• https://arxiv.org/abs/1711.09314
• https://arxiv.org/abs/1603.07929
• https://arxiv.org/abs/1504.03617
• https://arxiv.org/abs/1504.02683
• https://arxiv.org/abs/1103.3993

for non-Abelian symmetries and higher spin.

About the second question of yours just note that the following worldline action

$$ S=\int d\tau \left(\dot{x}^{\mu}p_{\mu}+l^{\mu}p_{\mu}\right) $$

with $\mu=0$ to $2$ reproduces Chern-Simons perturbativelly. (For more details see this).

About the first question of yours a good way to start is to realize that imposing the supersymmetry to be local leads to $p_{m}\gamma^{m}|\psi\rangle= m |\psi\rangle$, while local diffeomorphism leads to $p_{m}p^{m}|\psi\rangle=m^{2}|\psi\rangle$. If you want to describe a particle with more spinorial indices we need to add more supersymmetries and promote them to be local as well since each spinorial indice must obey the Dirac equation.

As an example let us construct a massless spin 1 particle in $d=10$. Following the reasoning above we must have two supersymmetries. In $d=10$ there is two ways of doing it, spinors with same chirality (Type IIB) or opposite chirality (Type IIA). This will lead to states:

$$ F^{\alpha}\,_{\beta}=\delta_{\alpha}^{\alpha}F+(\gamma^{mn})^{\alpha}\,_{\beta}F_{mn}+(\gamma^{mnpq})^{\alpha}\,_{\beta} F_{mnpq} $$

$$ F_{\alpha\beta}=\gamma^{m}_{\alpha\beta}F_{m}+\gamma^{mnp}_{\alpha\beta}F_{mnp}+\gamma_{\alpha\beta}^{mnpqr}F_{mnpqr} $$

and I will left as an exercise to check that the Dirac equation (setting $m=0$) for both spinorial indices implies that the $F$'s are field strengths of abelian gauge fields. These are the Ramond-Ramond fields that appear in type II superstring theories.

Another good example is a massless particle with spin $s$ in $d=4$. In this case we have something similar:

$$ (\sigma^{\mu})_{\dot\alpha}^{\alpha_1}\partial_{\mu} F_{\alpha_1\alpha_2\dots\alpha_{2s}}=0,\qquad \partial_{\mu}\partial^{\mu}F_{\alpha_1\alpha_2\dots\alpha_{2s}}=0 $$

the equations above describe the field strength of a massless particle of spin $s$, where $F_{\alpha_1\alpha_2\dots\alpha_{2s}}$ is totally symmetric. This can be generalized in an obvious way for other dimensions. From the worldline, these equations can be obtained from $2s$ local supersymmetries plus some additional local symmetries to get rid of unwanted worldline fields that will appear since worldline sumpersymmetry require equal number of bosonic and fermionic modes.