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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=-\int d\tau \sqrt{-\dot{x}^\mu \dot{x}_\mu} \end{equation}\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}} \end{equation}\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 \end{equation}\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 \end{equation}\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 $H=0$, $Q=0$.$$H=0, Q=0.\tag{5}$$

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 \end{equation}\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.

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=-\int d\tau \sqrt{-\dot{x}^\mu \dot{x}_\mu} \end{equation} and computing the momentum \begin{equation} p_\mu=\frac{m\dot{x}_\mu}{\sqrt{-\dot{x}^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 \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 \end{equation} i.e. the dynamics on the phase space should take place on hypersurfaces $H=0$, $Q=0$.

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 \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.

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.

3 There is a minus under the sqrt in the second equation which isn't in the first one. – DanielC
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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=\int d\tau \sqrt{\dot{x}^\mu \dot{x}_\mu} \end{equation}\begin{equation} S=-\int d\tau \sqrt{-\dot{x}^\mu \dot{x}_\mu} \end{equation} and computing the momentum \begin{equation} p_\mu=\frac{m\dot{x}_\mu}{\sqrt{-\dot{x}^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 \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 \end{equation} i.e. the dynamics on the phase space should take place on hypersurfaces $H=0$, $Q=0$.

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 \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 $N=1$${\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.

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=\int d\tau \sqrt{\dot{x}^\mu \dot{x}_\mu} \end{equation} and computing the momentum \begin{equation} p_\mu=\frac{m\dot{x}_\mu}{\sqrt{-\dot{x}^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 \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 \end{equation} i.e. the dynamics on the phase space should take place on hypersurfaces $H=0$, $Q=0$.

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 \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 $N=1$ 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.

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=-\int d\tau \sqrt{-\dot{x}^\mu \dot{x}_\mu} \end{equation} and computing the momentum \begin{equation} p_\mu=\frac{m\dot{x}_\mu}{\sqrt{-\dot{x}^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 \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 \end{equation} i.e. the dynamics on the phase space should take place on hypersurfaces $H=0$, $Q=0$.

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 \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.

2 added 502 characters in body; edited tags
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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=\int d\tau \sqrt{\dot{x}^\mu \dot{x}_\mu} \end{equation} and computing the momentum \begin{equation} p_\mu=\frac{m\dot{x}_\mu}{\sqrt{-\dot{x}^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 \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 \end{equation} i.e. the dynamics on the phase space should take place on hypersurfaces $H=0$, $Q=0$.

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 \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 $N=1$ 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.

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=\int d\tau \sqrt{\dot{x}^\mu \dot{x}_\mu} \end{equation} and computing the momentum \begin{equation} p_\mu=\frac{m\dot{x}_\mu}{\sqrt{-\dot{x}^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 \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 \end{equation} i.e. the dynamics on the phase space should take place on hypersurfaces $H=0$, $Q=0$.

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 \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 $Q$?

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=\int d\tau \sqrt{\dot{x}^\mu \dot{x}_\mu} \end{equation} and computing the momentum \begin{equation} p_\mu=\frac{m\dot{x}_\mu}{\sqrt{-\dot{x}^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 \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 \end{equation} i.e. the dynamics on the phase space should take place on hypersurfaces $H=0$, $Q=0$.

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 \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 $N=1$ 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.

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