Tweeted twitter.com/StackPhysics/status/1086005445918187522 occurred Jan 17 at 21:00 4 added 43 characters in body edited Jan 4 at 12:04 Qmechanic♦ 110k122101291 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 $$$$S=-\int d\tau \sqrt{-\dot{x}^\mu \dot{x}_\mu}$$$$$$$$S=-m\int d\tau \sqrt{-\dot{x}^\mu \dot{x}_\mu}\tag{1}$$$$ and computing the momentum $$$$p_\mu=\frac{m\dot{x}_\mu}{\sqrt{-\dot{x}^2}}$$$$$$$$p_\mu=\frac{m\dot{x}_\mu}{\sqrt{-\dot{x}^2}}\tag{2}$$$$ I see that it satisfies the constraint $$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$$, according to my notes. The action should take the form $$$$S=\int d\tau \space \dot{x}^\mu\dot{x}_\mu+\frac{i}{2}\psi_\mu\dot{\psi}^\mu$$$$$$$$S=\int d\tau \space \dot{x}^\mu\dot{x}_\mu+\frac{i}{2}\psi_\mu\dot{\psi}^\mu\tag{3}$$$$ which exhibits a supersymmetry on the worldline, the supersymmetric conserved charge being $$Q=\psi^\mu p_\mu$$. According to the lecturer I should find the constraints $$$$H=\frac{1}{2}p^2, \quad Q=\psi^\mu p_\mu$$$$$$$$H=\frac{1}{2}p^2, \quad Q=\psi^\mu p_\mu\tag{4}$$$$ i.e. the dynamics on the phase space should take place on hypersurfaces $$H=0$$, $$Q=0$$.$$H=0, Q=0.\tag{5}$$ My question: How can I derive these constraints? They should arise simply with the definition of momenta, but, having no constants to work with, I'm left with $$$$p_\mu=\dot{x}_\mu\\ \Pi_\mu=\frac{i}{2}\dot{\psi}_\mu$$$$$$$$p_\mu=\dot{x}_\mu\\ \Pi_\mu=\frac{i}{2}\dot{\psi}_\mu\tag{6}$$$$ and I don't know what to do with them. I see that, in principle, the first constraint is obtained by setting $$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$$ conserved charges. Is it the only way to find these constraints? Should I need this previous knowledge about supersymmetry to study the model? I think I should be able to find these constraints just by looking at the Lagrangian itself. I am trying to apply the Constrained Hamiltonian Systems theory on relativistic particles. For what concerns the scalar particle there is no issue. Indeed, I have the action $$$$S=-\int d\tau \sqrt{-\dot{x}^\mu \dot{x}_\mu}$$$$ and computing the momentum $$$$p_\mu=\frac{m\dot{x}_\mu}{\sqrt{-\dot{x}^2}}$$$$ I see that it satisfies the constraint $$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$$, according to my notes. The action should take the form $$$$S=\int d\tau \space \dot{x}^\mu\dot{x}_\mu+\frac{i}{2}\psi_\mu\dot{\psi}^\mu$$$$ which exhibits a supersymmetry on the worldline, the supersymmetric conserved charge being $$Q=\psi^\mu p_\mu$$. According to the lecturer I should find the constraints $$$$H=\frac{1}{2}p^2, \quad Q=\psi^\mu p_\mu$$$$ i.e. the dynamics on the phase space should take place on hypersurfaces $$H=0$$, $$Q=0$$. My question: How can I derive these constraints? They should arise simply with the definition of momenta, but, having no constants to work with, I'm left with $$$$p_\mu=\dot{x}_\mu\\ \Pi_\mu=\frac{i}{2}\dot{\psi}_\mu$$$$ and I don't know what to do with them. I see that, in principle, the first constraint is obtained by setting $$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$$ conserved charges. Is it the only way to find these constraints? Should I need this previous knowledge about supersymmetry to study the model? I think I should be able to find these constraints just by looking at the Lagrangian itself. I am trying to apply the Constrained Hamiltonian Systems theory on relativistic particles. For what concerns the scalar particle there is no issue. Indeed, I have the action $$$$S=-m\int d\tau \sqrt{-\dot{x}^\mu \dot{x}_\mu}\tag{1}$$$$ and computing the momentum $$$$p_\mu=\frac{m\dot{x}_\mu}{\sqrt{-\dot{x}^2}}\tag{2}$$$$ I see that it satisfies the constraint $$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$$, according to my notes. The action should take the form $$$$S=\int d\tau \space \dot{x}^\mu\dot{x}_\mu+\frac{i}{2}\psi_\mu\dot{\psi}^\mu\tag{3}$$$$ which exhibits a supersymmetry on the worldline, the supersymmetric conserved charge being $$Q=\psi^\mu p_\mu$$. According to the lecturer I should find the constraints $$$$H=\frac{1}{2}p^2, \quad Q=\psi^\mu p_\mu\tag{4}$$$$ i.e. the dynamics on the phase space should take place on hypersurfaces $$H=0, Q=0.\tag{5}$$ My question: How can I derive these constraints? They should arise simply with the definition of momenta, but, having no constants to work with, I'm left with $$$$p_\mu=\dot{x}_\mu\\ \Pi_\mu=\frac{i}{2}\dot{\psi}_\mu\tag{6}$$$$ and I don't know what to do with them. I see that, in principle, the first constraint is obtained by setting $$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$$ conserved charges. Is it the only way to find these constraints? Should I need this previous knowledge about supersymmetry to study the model? I think I should be able to find these constraints just by looking at the Lagrangian itself. 3 There is a minus under the sqrt in the second equation which isn't in the first one. – DanielC edited Jan 4 at 7:13 Qmechanic♦ 110k122101291 I am trying to apply the Constrained Hamiltonian Systems theory on relativistic particles. For what concerns the scalar particle there is no issue. Indeed, I have the action $$$$S=\int d\tau \sqrt{\dot{x}^\mu \dot{x}_\mu}$$$$$$$$S=-\int d\tau \sqrt{-\dot{x}^\mu \dot{x}_\mu}$$$$ and computing the momentum $$$$p_\mu=\frac{m\dot{x}_\mu}{\sqrt{-\dot{x}^2}}$$$$ I see that it satisfies the constraint $$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$$, according to my notes. The action should take the form $$$$S=\int d\tau \space \dot{x}^\mu\dot{x}_\mu+\frac{i}{2}\psi_\mu\dot{\psi}^\mu$$$$ which exhibits a supersymmetry on the worldline, the supersymmetric conserved charge being $$Q=\psi^\mu p_\mu$$. According to the lecturer I should find the constraints $$$$H=\frac{1}{2}p^2, \quad Q=\psi^\mu p_\mu$$$$ i.e. the dynamics on the phase space should take place on hypersurfaces $$H=0$$, $$Q=0$$. My question: How can I derive these constraints? They should arise simply with the definition of momenta, but, having no constants to work with, I'm left with $$$$p_\mu=\dot{x}_\mu\\ \Pi_\mu=\frac{i}{2}\dot{\psi}_\mu$$$$ and I don't know what to do with them. I see that, in principle, the first constraint is obtained by setting $$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 $$$$S=\int d\tau \sqrt{\dot{x}^\mu \dot{x}_\mu}$$$$ and computing the momentum $$$$p_\mu=\frac{m\dot{x}_\mu}{\sqrt{-\dot{x}^2}}$$$$ I see that it satisfies the constraint $$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$$, according to my notes. The action should take the form $$$$S=\int d\tau \space \dot{x}^\mu\dot{x}_\mu+\frac{i}{2}\psi_\mu\dot{\psi}^\mu$$$$ which exhibits a supersymmetry on the worldline, the supersymmetric conserved charge being $$Q=\psi^\mu p_\mu$$. According to the lecturer I should find the constraints $$$$H=\frac{1}{2}p^2, \quad Q=\psi^\mu p_\mu$$$$ i.e. the dynamics on the phase space should take place on hypersurfaces $$H=0$$, $$Q=0$$. My question: How can I derive these constraints? They should arise simply with the definition of momenta, but, having no constants to work with, I'm left with $$$$p_\mu=\dot{x}_\mu\\ \Pi_\mu=\frac{i}{2}\dot{\psi}_\mu$$$$ and I don't know what to do with them. I see that, in principle, the first constraint is obtained by setting $$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$$ conserved charges. Is it the only way to find these constraints? Should I need this previous knowledge about supersymmetry to study the model? I think I should be able to find these constraints just by looking at the Lagrangian itself. I am trying to apply the Constrained Hamiltonian Systems theory on relativistic particles. For what concerns the scalar particle there is no issue. Indeed, I have the action $$$$S=-\int d\tau \sqrt{-\dot{x}^\mu \dot{x}_\mu}$$$$ and computing the momentum $$$$p_\mu=\frac{m\dot{x}_\mu}{\sqrt{-\dot{x}^2}}$$$$ I see that it satisfies the constraint $$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$$, according to my notes. The action should take the form $$$$S=\int d\tau \space \dot{x}^\mu\dot{x}_\mu+\frac{i}{2}\psi_\mu\dot{\psi}^\mu$$$$ which exhibits a supersymmetry on the worldline, the supersymmetric conserved charge being $$Q=\psi^\mu p_\mu$$. According to the lecturer I should find the constraints $$$$H=\frac{1}{2}p^2, \quad Q=\psi^\mu p_\mu$$$$ i.e. the dynamics on the phase space should take place on hypersurfaces $$H=0$$, $$Q=0$$. My question: How can I derive these constraints? They should arise simply with the definition of momenta, but, having no constants to work with, I'm left with $$$$p_\mu=\dot{x}_\mu\\ \Pi_\mu=\frac{i}{2}\dot{\psi}_\mu$$$$ and I don't know what to do with them. I see that, in principle, the first constraint is obtained by setting $$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$$ conserved charges. Is it the only way to find these constraints? Should I need this previous knowledge about supersymmetry to study the model? I think I should be able to find these constraints just by looking at the Lagrangian itself. 2 added 502 characters in body; edited tags edited Jan 3 at 18:00 Luthien 952418 I am trying to apply the Constrained Hamiltonian Systems theory on relativistic particles. For what concerns the scalar particle there is no issue. Indeed, I have the action $$$$S=\int d\tau \sqrt{\dot{x}^\mu \dot{x}_\mu}$$$$ and computing the momentum $$$$p_\mu=\frac{m\dot{x}_\mu}{\sqrt{-\dot{x}^2}}$$$$ I see that it satisfies the constraint $$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$$, according to my notes. The action should take the form $$$$S=\int d\tau \space \dot{x}^\mu\dot{x}_\mu+\frac{i}{2}\psi_\mu\dot{\psi}^\mu$$$$ which exhibits a supersymmetry on the worldline, the supersymmetric conserved charge being $$Q=\psi^\mu p_\mu$$. According to the lecturer I should find the constraints $$$$H=\frac{1}{2}p^2, \quad Q=\psi^\mu p_\mu$$$$ i.e. the dynamics on the phase space should take place on hypersurfaces $$H=0$$, $$Q=0$$. My question: How can I derive these constraints? They should arise simply with the definition of momenta, but, having no constants to work with, I'm left with $$$$p_\mu=\dot{x}_\mu\\ \Pi_\mu=\frac{i}{2}\dot{\psi}_\mu$$$$ and I don't know what to do with them. I see that, in principle, the first constraint is obtained by setting $$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$$ conserved charges. Is it the only way to find these constraints? Should I need this previous knowledge about supersymmetry to study the model? I think I should be able to find these constraints just by looking at the Lagrangian itself. I am trying to apply the Constrained Hamiltonian Systems theory on relativistic particles. For what concerns the scalar particle there is no issue. Indeed, I have the action $$$$S=\int d\tau \sqrt{\dot{x}^\mu \dot{x}_\mu}$$$$ and computing the momentum $$$$p_\mu=\frac{m\dot{x}_\mu}{\sqrt{-\dot{x}^2}}$$$$ I see that it satisfies the constraint $$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$$, according to my notes. The action should take the form $$$$S=\int d\tau \space \dot{x}^\mu\dot{x}_\mu+\frac{i}{2}\psi_\mu\dot{\psi}^\mu$$$$ which exhibits a supersymmetry on the worldline, the supersymmetric conserved charge being $$Q=\psi^\mu p_\mu$$. According to the lecturer I should find the constraints $$$$H=\frac{1}{2}p^2, \quad Q=\psi^\mu p_\mu$$$$ i.e. the dynamics on the phase space should take place on hypersurfaces $$H=0$$, $$Q=0$$. My question: How can I derive these constraints? They should arise simply with the definition of momenta, but, having no constants to work with, I'm left with $$$$p_\mu=\dot{x}_\mu\\ \Pi_\mu=\frac{i}{2}\dot{\psi}_\mu$$$$ and I don't know what to do with them. I see that, in principle, the first constraint is obtained by setting $$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$$? I am trying to apply the Constrained Hamiltonian Systems theory on relativistic particles. For what concerns the scalar particle there is no issue. Indeed, I have the action $$$$S=\int d\tau \sqrt{\dot{x}^\mu \dot{x}_\mu}$$$$ and computing the momentum $$$$p_\mu=\frac{m\dot{x}_\mu}{\sqrt{-\dot{x}^2}}$$$$ I see that it satisfies the constraint $$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$$, according to my notes. The action should take the form $$$$S=\int d\tau \space \dot{x}^\mu\dot{x}_\mu+\frac{i}{2}\psi_\mu\dot{\psi}^\mu$$$$ which exhibits a supersymmetry on the worldline, the supersymmetric conserved charge being $$Q=\psi^\mu p_\mu$$. According to the lecturer I should find the constraints $$$$H=\frac{1}{2}p^2, \quad Q=\psi^\mu p_\mu$$$$ i.e. the dynamics on the phase space should take place on hypersurfaces $$H=0$$, $$Q=0$$. My question: How can I derive these constraints? They should arise simply with the definition of momenta, but, having no constants to work with, I'm left with $$$$p_\mu=\dot{x}_\mu\\ \Pi_\mu=\frac{i}{2}\dot{\psi}_\mu$$$$ and I don't know what to do with them. I see that, in principle, the first constraint is obtained by setting $$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. 1 asked Jan 3 at 17:37 Luthien 952418