Spinning Particles in Background Gauge Fields

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?

• 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. – Qmechanic Jan 16 at 17:39
• @Qmechanic Thank you very much! I will correct it. – The Last Knight of Silk Road Jan 16 at 17:40
• @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? – Mozibur Ullah Jan 16 at 17:56
• Yeah, they're my abbreviations in various Phys.SE posts :) – Qmechanic Jan 16 at 17:58

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 $$c=1$$.

• Thank you. Do you know any textbook where I can find the expression of its Lagrangian? – The Last Knight of Silk Road Jan 13 at 18:08
• More references: 2. L. Brink, P. Di Vecchia & P. Howe, Nucl. Phys. B118 (1977) 76; eq. (5.8). – Qmechanic Jan 14 at 18:01
• 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? – The Last Knight of Silk Road Jan 14 at 18:21