Constrained Hamiltonian systems: spin 1/2 particle

Question

I am trying to apply the Constrained Hamiltonian Systems theory on relativistic particles. For what concerns the scalar particle there is no issue. Indeed, I have the action \begin{equation} S=-m\int d\tau \sqrt{-\dot{x}^\mu \dot{x}_\mu}\tag{1} \end{equation} and computing the momentum \begin{equation} p_\mu=\frac{m\dot{x}_\mu}{\sqrt{-\dot{x}^2}}\tag{2} \end{equation} I see that it satisfies the constraint $p_\mu p^\mu+m^2=0$. I then proceed to quantize the system with the Dirac method.

I am finding issues with the relativistic massless spin 1/2 particle. Indeed, it is described by the space-time coordinates $x^\mu$ and by the real grassmann variables $\psi^\mu$, according to my notes. The action should take the form \begin{equation} S=\int d\tau \space \dot{x}^\mu\dot{x}_\mu+\frac{i}{2}\psi_\mu\dot{\psi}^\mu\tag{3} \end{equation} which exhibits a supersymmetry on the worldline, the supersymmetric conserved charge being $Q=\psi^\mu p_\mu$. According to the lecturer I should find the constraints \begin{equation} H=\frac{1}{2}p^2, \quad Q=\psi^\mu p_\mu\tag{4} \end{equation} i.e. the dynamics on the phase space should take place on hypersurfaces $$H=0, Q=0.\tag{5}$$

My question: How can I derive these constraints? They should arise simply with the definition of momenta, but, having no constants to work with, I'm left with \begin{equation} p_\mu=\dot{x}_\mu\\ \Pi_\mu=\frac{i}{2}\dot{\psi}_\mu\tag{6} \end{equation} and I don't know what to do with them. I see that, in principle, the first constraint is obtained by setting $m=0$ in the constraint of the scalar particle for example, but what if I want to derive it without the previous knowledge? And what about $Q$?

Edit: By intuition, knowing that the model exhibits a ${\cal N}=1$ supersymmetry, I may understand that the dynamics must take place on a surface such that $H=const$ and $Q=const$ (then I could set the constant to zero without lack of generality?), being $Q$ and $H$ conserved charges. Is it the only way to find these constraints? Should I need this previous knowledge about supersymmetry to study the model? I think I should be able to find these constraints just by looking at the Lagrangian itself.

Answer 1

1. We consider here the massless case $m=0$. Let us start from the Lagrangian$^1$ $$L_0~=~\frac{\dot{x}^2}{2e} +\frac{i}{2}\psi_{\mu}\dot{\psi}^{\mu} \tag{A}$$ with an einbein field $e$, cf. e.g. this Phys.SE post. If we introduce the momentum $$ p_{\mu}~=~\frac{\partial L_0}{\partial \dot{x}^{\mu}} ~=~\frac{\dot{x}_{\mu}}{e},\tag{B}$$ the corresponding Legendre transformation $\dot{x}^{\mu}\leftrightarrow p_{\mu}$ yields a first-order Lagrangian $$\begin{align} L_1~=~&p_{\mu}\dot{x}^{\mu} +\frac{i}{2}\psi_{\mu}\dot{\psi}^{\mu}-eH, \cr H~:=~&\frac{p^2}{2}.\end{align} \tag{C}$$ This explains OP's first constraint $H\approx 0$, which is indirectly due to world-line (WL) reparametrization invariance, cf. this Phys.SE post.

2. It is unnecessary to introduce momentum for the fermions $\psi^{\mu}$ as the Lagrangian $L_1$ is already on first-order form, cf. the Faddeev-Jackiw method.

3. The Lagrangian $L_1$ has a global super quasisymmetry. The infinitesimal transformation $$\begin{align} \delta x^{\mu}~=~&i\varepsilon\psi^{\mu}, \cr \delta \psi^{\mu}~=~&-\varepsilon p^{\mu}, \cr \delta p^{\mu}~=~& 0 , \cr \delta e~=~& 0, \end{align} \tag{D}$$ changes the Lagrangian with a total derivative $$\begin{align} \delta L_1~=~&\ldots~=~i\dot{\varepsilon}Q+\frac{i}{2}\frac{d(\varepsilon Q)}{d\tau}, \cr Q~:=~&p_{\mu}\psi^{\mu}, \end{align}\tag{E}$$ for $\tau$-independent Grassmann-odd infinitesimal parameter $\varepsilon$.

4. OP's other constraint $Q\approx0$ arises by gauging the SUSY, i.e. $\delta L_1$ should be a total derivative for an arbitrary function $\varepsilon(\tau)$. On reason to do this is given in Ref. 2 below eq. (3.3):

   Because of the time component of the field $\psi^{\mu}$ there is a possibility that negative norm states may appear in the physical spectrum. In order to decouple them we require an additional invariance and, inspired by the Neveu-Schwarz-Ramond model, it seems natural to demand invariance under local supergauge transformations.

5. Concretely, we impose $Q\approx0$ with the help of a Lagrange multiplier $\chi$. This leads to the Lagrangian $$\begin{align} L_2~=~&L_1-i \chi Q\cr ~=~&p_{\mu}\dot{x}^{\mu} +\frac{i}{2}\psi_{\mu}\dot{\psi}^{\mu}-eH - i \chi Q .\end{align}\tag{F}$$

6. Let us mention for completeness that in order to have gauged super quasisymmetry of the new Lagrangian $L_2$, the previous transformation $\delta e= 0$ needs to be modified into $$\begin{align} \delta e~=~&2i\chi\varepsilon, \cr \delta \chi~=~&\dot{\varepsilon}.\end{align}\tag{G}$$

7. An alternative perspective is the replacement $$L_2~=~ L_1|_{\dot{x}\to Dx}\tag{H}$$ of the ordinary derivative $$\dot{x}^{\mu}\quad\longrightarrow\quad Dx^{\mu}~:=~\dot{x}^{\mu} -i\chi \psi^{\mu}\tag{I}$$ with a gauge-covariant derivative $Dx^{\mu}$. Here $\chi$ is a compensating gauge field. The gauge-covariant derivative transforms as $$ \delta Dx^{\mu}~=~i\varepsilon(\dot{\psi}^{\mu}-\chi p^{\mu}).\tag{J}$$

References:

1. F. Bastianelli, Constrained hamiltonian systems and relativistic particles, 2017 lecture notes; Section 2.2.

2. L. Brink, P. Di Vecchia & P. Howe, Nucl. Phys. B118 (1977) 76; Below eq. (3.3).

3. C.M. Hull & J.-L. Vazquez-Bello, arXiv:hep-th/9308022; Chapter 2, p. 7-8.

4. O. Corradini & C. Schubert, Spinning Particles in QM & QFT, arXiv:1512.08694; Subsection 1.5.2.

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