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=-mint dtau sqrt{-dot{x}^mu dot{x}_mu}tag{1}
end{equation}
and computing the momentum 
begin{equation}
p_mu=frac{mdot{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 dtau space dot{x}^mudot{x}_mu+frac{i}{2}psi_mudot{psi}^mutag{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_mutag{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}_mutag{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.

Qmechanic · Accepted Answer

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 $Happrox 0$, which is indirectly due to world-line (WL) reparametrization invariance, cf. this Phys.SE post.

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.

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

OP's other constraint $Qapprox0$ 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.

Concretely, we impose $Qapprox0$ with the help of a Lagrange multiplier $chi$. This leads to the Lagrangian
$$begin{align} L_2~=~&L_1-i chi Qcr
~=~&p_{mu}dot{x}^{mu} +frac{i}{2}psi_{mu}dot{psi}^{mu}-eH - i chi Q .end{align}tag{F}$$

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
$$begin{align} delta e~=~&2ichivarepsilon, cr
 delta chi~=~&dot{varepsilon}.end{align}tag{G}$$

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

References:

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

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

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

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

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

Constrained Hamiltonian systems: spin 1/2 particle

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