1

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?

Valac
  • 2,893

1 Answers1

1

This e.g. explained in Ref. 1.

  1. The massive spin 1/2 particle without an EM background is described by a Hamiltonian Lagrangian $$ \begin{align}L_H~=~&p_{\mu}\dot{x}^{\mu} +\frac{i}{2}(\psi_{\mu}\dot{\psi}^{\mu}+\psi_5\dot{\psi}^5)\cr &-eH - i \chi Q, \cr H~:=~&\frac{1}{2}(p^2+m^2), \cr Q~:=~&p_{\mu}\psi^{\mu}+m\psi^5 .\end{align}\tag{80}$$ 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 $$ \begin{align} H~:=~&\frac{1}{2}((p-qA)^2+m^2)+\frac{iq}{2}F_{\mu\nu}\psi^{\mu}\psi^{\nu}, \cr Q~:=~&(p_{\mu}-qA_{\mu})\psi^{\mu}+m\psi^5 , \end{align} $$ 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.

  2. L. Brink, P. Di Vecchia & P. Howe, Nucl. Phys. B118 (1977) 76; eq. (5.8).

  3. O. Corradini & C. Schubert, Spinning Particles in QM & QFT, arXiv:1512.08694; Section 1.5, p. 41.

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

Qmechanic
  • 201,751
  • Notes for later: ${\cal N}=1$ spinning particle. Lagr. $\quad L =\frac{\dot{x}^2}{2e} -\frac{i}{e}\chi\psi_{\mu}\dot{x}^{\mu} +\frac{i}{2}\psi_a \frac{D\psi^a}{D\tau} +\frac{i}{2}\psi_5\dot{\psi}^5 -\frac{e}{2}m^2 -im\chi\psi^5 $ $+q\dot{x}^{\mu}A_{\mu} -\frac{iqe}{2}F_{\mu\nu}\psi^{\mu}\psi^{\nu}$; https://physics.stackexchange.com/q/114862 – Qmechanic Jan 12 '20 at 17:41
  • $\quad\psi^{\mu} = E^{\mu}{}a\psi^a$; $\quad{\psi^a,\psi^b}{PB} =(-i)\eta^{ab}$; $\quad\frac{D\psi^a}{D\tau} =\dot{\psi}^a +\dot{x}^{\mu}\omega_{\mu}{}^a{}b\psi^b$; $\quad 0 =(\nabla{\mu}e)^{a}{}{\nu} =\partial{\mu}e^{a}{}{\nu} +\omega{\mu}{}^a{}b e^b{}{\nu}- e^a{}{\lambda}\Gamma{\mu\nu}^{\lambda}$; https://physics.stackexchange.com/q/119838 $\quad\frac{Dx^{\mu}}{D\tau} =\dot{x}^{\mu}$ is more like a definition; does not really make sense. – Qmechanic Jan 21 '23 at 14:29
  • Can. mom. $\quad p_{\mu}= \frac{\partial L}{\partial \dot{x}^{\mu}}= \frac{1}{e}\dot{x}{\mu} -\frac{i}{e} \chi \psi{\mu} +\frac{i}{2}\omega_{\mu ab}\psi^a\psi^b +qA_{\mu}$; Kin. mom. $\quad \pi_{\mu}=p_{\mu}-qA_{\mu}-\frac{i}{2}\omega_{\mu ab}\psi^a\psi^b$; Ham. $\quad H = p_{\mu}\dot{x}^{\mu}+\frac{i}{2}\psi_a \dot{\psi}^a +\frac{i}{2}\psi_5\dot{\psi}^5 -L =\frac{\dot{x}^2}{2e} +\frac{e}{2}m^2 +im\chi\psi^5+\frac{iqe}{2} F_{\mu\nu}\psi^{\mu}\psi^{\nu}$ $= \frac{e}{2}\left(\pi +\frac{i}{e}\chi\psi \right)^2 + \ldots$ $=eT+i\chi\Theta$; – Qmechanic Jan 21 '23 at 14:41
  • SUSY charge: $\quad\Theta =\pi_{\mu}\psi^{\mu} +m\psi^5$; $\quad T =\frac{1}{2} (\pi^2+m^2) +\frac{iq}{2} F_{\mu\nu}\psi^{\mu}\psi^{\nu}$; $\quad {\pi_{\mu},\pi_{\nu}}{PB}^{ET} =\frac{i}{2}R{\mu\nu ab}\psi^a\psi^b +qF_{\mu\nu}$; $\quad {\pi_{\mu},\psi^{\lambda}}{PB}^{ET} =\Gamma^{\lambda}{\mu\nu}\psi^{\nu}$; $\quad{\Theta,\Theta}{PB}^{ET} = \pi{\mu}{\psi^{\mu},\psi^{\nu}}{PB}\pi{\nu} +qF_{\mu\nu}\psi^{\mu}\psi^{\nu} +(-i)m^2 =2(-i)T$; The kinetic spin connection term is crucial. It does not work with the Christoffel connection, i.e. the flat index on $\psi^a$ is crucial. – Qmechanic Jan 21 '23 at 15:23
  • Ham. Lagr. $\quad L_H= p_{\mu}\dot{x}^{\mu} +\frac{i}{2}\psi_a \dot{\psi}^a +\frac{i}{2}\psi_5\dot{\psi}^5 - H$; BRST transf. $\quad {\bf s}={\mathbb{Q},\cdot}_{PB}$; $\quad\mathbb{Q}=\int!\rho~d\tau~Q$; $\quad -Q=TC+\Theta\gamma+\epsilon BP+i\epsilon\beta\pi$; – Qmechanic Jan 21 '23 at 15:35
  • $\quad {\bf s}x^{\mu} =C\pi^{\mu}+\gamma\psi^{\mu}$; $\quad {\bf s}p_{\mu}={p_{\mu},T}C+\ldots=-\frac{1}{2}\partial_{\mu}g^{\nu\lambda}\pi_{\nu}\pi_{\lambda}C +q\partial_{\mu}A_{\nu}\pi^{\nu}C+\ldots$; $\quad {\bf s}\pi_{\mu}={\pi_{\mu},T}C+\ldots =-\frac{1}{2}\partial_{\mu}g^{\nu\lambda}\pi_{\nu}\pi_{\lambda}C+qF_{\mu\nu}\pi^{\nu}C+\ldots$; $\quad {\bf s}e=\rho P\approx \dot{C}$; $\quad {\bf s}\chi =-i\rho\pi\approx-i\dot{\gamma}$; $\quad {\bf s}\psi^a=\gamma(i\pi^a+\psi^{\mu}\omega_{\mu b}{}^a\psi^{b})+C(\pi^{\mu}\omega_{\mu b}{}^a\psi^b-q\psi_b F^{ba})$; $\quad {\bf s}\psi^5=im\gamma$; – Qmechanic Jan 21 '23 at 16:47
  • Superfield formalism: $\quad X^{\mu} =x^{\mu} +i\theta\sqrt{e} \psi^{\mu}$;$\quad X^5 =x^5 +i\theta\sqrt{e} \psi^5$; $\quad D =i\frac{\partial}{\partial\theta} +\theta\frac{\partial}{\partial\tau}$; $\quad D^2=i\frac{\partial}{\partial\tau}$; $\quad DX^{\mu} =\theta\dot{x}^{\mu} -\sqrt{e}\psi^{\mu}$; $\quad q\int!d\theta~A_{\mu}(X)DX^{\mu} =q\dot{x}^{\mu}A_{\mu} -\frac{iqe}{2}F_{\mu\nu}\psi^{\mu}\psi^{\nu}$; $\quad \frac{DX^{\mu}}{D\tau} =\dot{x}^{\mu} +i\theta \sqrt{e}E^{\mu}{}_a \frac{D\psi^a}{D\tau}+\ldots$; – Qmechanic Jan 22 '23 at 13:14
  • $\quad E =e -2i\theta \sqrt{e}\chi =e(1-2i\frac{\theta\chi}{\sqrt{e}})$; $\quad \int!d\theta~\frac{1}{2E} DX^{\mu} g_{\mu\nu}\frac{DX^{\nu}}{D\tau} =\frac{\dot{x}^2}{2e} -\frac{i}{e}\chi\psi_{\mu}\dot{x}^{\mu} +\frac{i}{2}\psi_a \frac{D\psi^a}{D\tau}$; – Qmechanic Jan 22 '23 at 15:07
  • $\quad P_{\mu} =p_{\mu} +\frac{i}{2}\frac{\theta}{\sqrt{e}}\dot{\psi}a e^a{}{\mu}$; $\quad\int!d\theta~P_{\mu} DX^{\mu} =p_{\mu}\dot{x}^{\mu} +\frac{i}{2}\psi_a \dot{\psi}^a =\int!d\theta~\Psi_{\mu}\dot{X}^{\mu}$; $\quad\Psi_{\mu} =-\frac{1}{2\sqrt{e}}\psi_{\mu} +\theta p_{\mu}$; $\quad\Theta =-\frac{1}{2\sqrt{e}}\Theta +\theta T$; $\quad {\cal C}=C+2\theta \sqrt{e}\gamma$; $\quad{\cal P}=P+2\theta \sqrt{e}\pi$; $\quad\bar{\pi} =\theta\bar{P}+\frac{1}{2\sqrt{e}}\bar{\pi}$; $\quad\bar{\gamma} =\theta\bar{C}+\frac{1}{2\sqrt{e}}\bar{\gamma}$; $\quad\beta =-\frac{i}{2\sqrt{e}}\beta+\theta B$; – Qmechanic Jan 22 '23 at 16:49
  • Ultra-local Poisson bracket: $\quad {x^{\mu}(\tau),p_{\nu}(\tau^{\prime})}{PB} =\delta^{\mu}{\nu}\frac{1}{\rho(\tau)}\delta(\tau!-!\tau^{\prime})$; $\quad{\psi^a(\tau),\psi^b(\tau^{\prime})}{PB} =(-i)\eta^{ab}\frac{1}{\rho(\tau)}\delta(\tau!-!\tau^{\prime})$; $\quad {C(\tau), \bar{P}(\tau^{\prime})}{PB}=\frac{1}{\rho(\tau)}\delta(\tau!-!\tau^{\prime})$; $\quad {\gamma(\tau), \bar{\pi}(\tau^{\prime})}_{PB}=\frac{1}{\rho(\tau)}\delta(\tau!-!\tau^{\prime})$; – Qmechanic Jan 23 '23 at 11:35
  • $\quad{\frac{e(\tau)}{\rho(\tau)},B(\tau^{\prime})}{PB} =\frac{1}{\epsilon\rho(\tau)}\delta(\tau!-!\tau^{\prime})$; $\quad{\frac{\chi(\tau)}{\rho(\tau)},\beta(\tau^{\prime})}{PB}=\frac{1}{\epsilon\rho(\tau)}\delta(\tau!-!\tau^{\prime})$; $\quad{P(\tau),\bar{C}(\tau^{\prime})}{PB}=\frac{1}{\epsilon\rho(\tau)}\delta(\tau!-!\tau^{\prime})$; $\quad{\pi(\tau),\bar{\gamma}(\tau^{\prime})}{PB}=\frac{1}{\epsilon\rho(\tau)}\delta(\tau!-!\tau^{\prime})$; Barred variables effectively serve as momenta. The Grassmann-odd effective momenta $\beta$, $\bar{C}$ and $\bar{P}$ are imaginary. – Qmechanic Jan 23 '23 at 11:47
  • $\quad \psi :=\int ! \mathrm{d}\tau\left(\bar{C}\left(\frac{\xi\rho}{2}B +G(e) +\epsilon \frac{d(e/\rho)}{d\tau}\right) -i\bar{\gamma}\left(G(\chi)+\epsilon\frac{d(\chi/\rho)}{d\tau}\right) -\bar{P}e +i\bar{\pi}\chi\right)$; $\quad S_{BFV} = \int ! \mathrm{d}\tau\left(p_{\mu}\dot{x}^{\mu} +\frac{i}{2}\psi_a \dot{\psi}^a +\frac{i}{2}\psi_5\dot{\psi}^5 +C\dot{\bar{P}} +\gamma\dot{\bar{\pi}}\right) -\left{ \psi, \mathbb{Q} \right}{PB} = \int ! \mathrm{d}\tau ~L{BFV}$; – Qmechanic Jan 23 '23 at 12:04
  • $\quad L_{BFV} =L_H +\epsilon\left( B \frac{d(e/\rho)}{d\tau} +\beta\frac{d(\chi/\rho)}{d\tau} + \bar{C} \dot{P}+\bar{\gamma}\dot{\pi}\right) +\bar{P}\left( \dot{C}-\rho P\right) + \bar{C} \rho G^{\prime} P + B \left(\frac{\xi\rho}{2}B+G(e)\right) +\bar{\pi}\left( \dot{\gamma}-\rho \pi\right) + \bar{\gamma} \rho G^{\prime} \pi + \beta G(\chi)$; $\quad L_{BV} =\ldots $; $\quad L_{\rm gf} =L_H +\left(\chi^{\prime}\bar{C}-\frac{\epsilon}{\rho}\dot{\bar{C}}\right)\dot{C} +B \left(\frac{\xi\rho}{2}B +\chi(e) +\epsilon \frac{d(e/\rho)}{d\tau}\right) +\ldots $; – Qmechanic Jan 23 '23 at 12:34