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A simple model for a spinning particle is

$$L=m\int dt\left(\dot{x}^{2}-\frac{i}{2}\psi\dot{\psi}\right)$$

with SUSY algebra $\delta x=-i\epsilon\psi$ and $\delta\psi=-\epsilon\dot{x}$, where $\epsilon$ is a Grassmann number.

I understand that the Lagrangian for a bosonic particle coupled with a background gauge field $A$ is

$$L=m\int dt\dot{x}^{2}+i\int dt\dot{x}^{\mu}A_{\mu}.$$

What is the action for a spinning particle coupled with the gauge field?

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  • $\begingroup$ Minor semantic comment for the record: A particle with world-line (WL) SUSY (as is the case here) is called a spinning particle in the literature, while the word superparticle is reserved for a particle with target space (TS) SUSY. $\endgroup$ – Qmechanic Jan 16 at 17:39
  • $\begingroup$ @Qmechanic Thank you very much! I will correct it. $\endgroup$ – Libertarian Monarchist Bot Jan 16 at 17:40
  • $\begingroup$ @Qmechanic: I don't think I've come across the abbreviation WL for worldline, and TS for targetspace, in my reading; they're not your invention by any chance? $\endgroup$ – Mozibur Ullah Jan 16 at 17:56
  • $\begingroup$ Yeah, they're my abbreviations in various Phys.SE posts :) $\endgroup$ – Qmechanic Jan 16 at 17:58
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This e.g. explained in Ref. 1.

  1. The massive spin 1/2 particle without an EM background is described by a Hamiltonian Lagrangian $$ L_H~=~p_{\mu}\dot{x}^{\mu} +\frac{i}{2}(\psi_{\mu}\dot{\psi}^{\mu}+\psi_5\dot{\psi}^5)-eH - i \chi Q, \qquad H~:=~p^2+m^2, \qquad Q~:=~p_{\mu}\psi^{\mu}+m\psi^5 .\tag{80}$$ For the massless case $m=0$, see also this related Phys.SE post.

  2. In an EM background, the Hamiltonian $H$ and supercharge $Q$ change to $$ H~:=~(p-qA)^2+m^2+\frac{iq}{2}F_{\mu\nu}\psi^{\mu}\psi^{\nu}, \qquad Q~:=~(p_{\mu}-qA_{\mu})\psi^{\mu}+m\psi^5 ,$$ cf. eq. (122) in Ref. 1.

  3. It would take us too far to try to explain every aspect of the above construction, but let us just briefly mention that it is possible perform an Legendre transformation to a Lagrangian formulation, and to gauge-fix the einbein field $e$, to achieve an action closer to OP's starting point.

References:

  1. F. Bastianelli, Constrained hamiltonian systems and relativistic particles, 2017 lecture notes; Section 2.3 + Chapter 3.

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$^1$ Conventions: We use the Minkowski sign convention $(-,+,+,+)$ and we work in units where $c=1$.

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  • $\begingroup$ Thank you. Do you know any textbook where I can find the expression of its Lagrangian? $\endgroup$ – Libertarian Monarchist Bot Jan 13 at 18:08
  • $\begingroup$ More references: 2. L. Brink, P. Di Vecchia & P. Howe, Nucl. Phys. B118 (1977) 76; eq. (5.8). $\endgroup$ – Qmechanic Jan 14 at 18:01
  • $\begingroup$ Thank you. I have a question. In equation 6.1 of reference 2, the $\gamma^{5}$ matrix exists only in even dimensions. Do you know where I can find the expression for a particle living in odd dimensions? $\endgroup$ – Libertarian Monarchist Bot Jan 14 at 18:21

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