Q: In the unconstrained relativistic point particle theory, what is the canonical momentum $$p_\mu$$ conjugate to $$X^\mu$$ for the action $$L = m \sqrt{\dot{X}^\nu \dot{X}_\nu}$$?

$$p_\mu = m \dot{X}_\mu / (\dot{X}^\nu \dot{X}_\nu$$)^{1/2}$$

Q: How does the parameter $$\Lambda$$ transform under a reparametrization $$\tau \to \tau'$$?

$$\Lambda' = \frac{d\tau}{d\tau'} \Lambda$$

Q: For the unconstrained action, what is the general solution for $$p^\mu in $$terms of $$M$$ and $$\dot{X}^\mu$$?

$$p^\mu = \frac{M \dot{X}^\mu}{(\dot{X}^\nu \dot{X}_\nu$$)^{1/2}$$}$$

Q: What is the equation of motion for the kinetic momentum $$\pi^\mu in $$the unconstrained theory?

$$\dot{\pi}^\mu \pi_\mu = \frac{1}{2} \frac{d}{d\tau}(\pi^\mu \pi_\mu) = 0$$

Question 1

In the unconstrained relativistic point particle theory, what is the canonical momentum $$p_\mu$$ conjugate to $$X^\mu$$ for the action $$L = m \sqrt{\dot{X}^
u \dot{X}_
u}$$?

Accepted Answer

$$p_\mu = m \dot{X}_\mu / (\dot{X}^
u \dot{X}_
u$$)^{1/2}$$

Question 2

Given the unconstrained action $$L = (1/2\Lambda) \dot{X}^\mu \dot{X}_\mu$$, what is the canonical momentum $$p_\mu$$?

Accepted Answer

$$p_\mu = \dot{X}_\mu / \Lambda$$

Question 3

For the unconstrained action $$L = (1/2\Lambda) \dot{X}^\mu \dot{X}_\mu$$, what is the mass-shell condition in terms of $$p^\mu$$ and $$\Lambda$$?

Accepted Answer

$$p^\mu p_\mu = \dot{X}^\mu \dot{X}_\mu / \$$Lambda^2$$

Question 4

How does the parameter $$\Lambda$$ transform under a reparametrization $$	au 	o 	au'$$?

Accepted Answer

$$\Lambda' = \frac{d	au}{d	au'} \Lambda$$

Question 5

For the unconstrained action, what is the general solution for $$p^\mu in $$terms of $$M$$ and $$\dot{X}^\mu$$?

Accepted Answer

$$p^\mu = \frac{M \dot{X}^\mu}{(\dot{X}^
u \dot{X}_
u$$)^{1/2}$$}$$

Question 6

What is the equation of motion for $$X^\mu in $$the unconstrained theory?

Accepted Answer

$$\dot{p}_\mu = 0$$

Question 7

What is the equation of motion for $$p_\mu in $$the unconstrained theory with $$L = (1/2\Lambda) \dot{X}^\mu \dot{X}_\mu$$?

Accepted Answer

$$p_\mu = \dot{X}_\mu / \Lambda$$

Question 8

Under a gauge transformation $$\phi(	au, x)$$, how do the gauge fields $$A_\mu$$ and $$V$$ transform?

Accepted Answer

$$eA'_\mu = eA_\mu + \partial_\mu \phi$$, $$eV' = eV + \partial \phi / \partial 	au$$

Question 9

For a particle in a background field, what is the condition for the kinetic momentum $$\pi^\mu to be a $$constant of motion?

Accepted Answer

$$\partial A_\mu / \partial s = 0$$, $$\partial_\mu V = 0$$

Question 10

What is the equation of motion for the kinetic momentum $$\pi^\mu in $$the unconstrained theory?

Accepted Answer

$$\dot{\pi}^\mu \pi_\mu = \frac{1}{2} \frac{d}{d	au}(\pi^\mu \pi_\mu) = 0$$

Question 11

What is the general solution for $$\pi^\mu in $$the unconstrained theory?

Accepted Answer

$$\pi^\mu = \frac{M \dot{X}^\mu}{(\dot{X}^\alpha \dot{X}_\$$alpha)^{1/2}$$}$$

Question 12

How does the canonical momentum $$p'_\mu$$ transform under a point transformation $$x^\mu 	o x'^\mu$$?

Accepted Answer

$$p'_\mu = \frac{\partial x^
u}{\partial x'^\mu} p_
u$$

Question 13

What is the Hamiltonian for the unconstrained action $$L = (1/2\Lambda)(p^\mu p_\mu - \$$kappa^2$$)$$?

Accepted Answer

$$H = \frac{\Lambda}{2}(p^\mu p_\mu - \$$kappa^2$$)$$

Question 14

What is the continuity equation for the probability density $$\rho$$ and current $$j^\mu in $$the unconstrained theory?

Accepted Answer

$$\frac{\partial ho}{\partial 	au} + D_\mu j^\mu = 0$$

Question 15

What is the Hamilton–Jacobi equation for the action $$S in $$the unconstrained theory?

Accepted Answer

$$-\frac{\partial S}{\partial 	au} = \frac{\Lambda}{2}(\partial_\mu S \partial^\mu S - \$$kappa^2$$)$$

Question 16

What is the normalization condition for the probability density $$A^2 in $$curved spacetime?

Accepted Answer

$$\int \sqrt{|g|} dx\, A^2$ = 1$$

Question 17

For a wave packet localized at $$x_c$$, what is the form of $$A^2$$?

Accepted Answer

$$A^2$$ = \delta(x - X_c) / \sqrt{|g|}$$

Question 18

What is the conservation law for the stress–energy tensor $$T^\mu_
u$$?

Accepted Answer

$$\oint d\Sigma_\mu T^\mu_
u \delta x^
u = 0$$

Question 19

What is the generator $$G(\Sigma)$$ for translations in spacetime?

Accepted Answer

$$G(\Sigma) = \int d\Sigma_\mu T^\mu_
u \delta x^
u$$

Question 20

What is the evolution equation for the wave function $$\psi in $$the Heisenberg picture?

Accepted Answer

$$\frac{\partial \psi}{\partial 	au} = -\{\psi, H\}$$

Question 21

What is the expression for the charge operator $$Q in $$terms of the current $$j_c^\mu$$?

Accepted Answer

$$Q = \int d\Sigma_\mu j_c^\mu$$

Question 22

What is the expression for the conserved charge $$C in $$terms of the creation and annihilation operators $$c^\dagger(p)$$ and $$c(p)$$?

Accepted Answer

$$C = -\alpha \int d^D p\, c^\dagger(p) c(p)$$

Question 23

What is the relation between $$\Lambda$$ and the proper time $$s_i$$ and $$p_i^2$ for a particle?

Accepted Answer

$$\Lambda = \frac{ds_i}{d	au} \frac{1}{\sqrt{p_i^2$}}$$

Question 24

What is the evolution equation for a field $$\phi in $$terms of the Hamiltonian $$P^0$$?

Accepted Answer

$$\frac{\partial \phi}{\partial x^0$} = i [\psi, P^0$]$$

Question 25

What is the velocity vector $$dx/d	au in $$terms of the basis vectors $$e_\mu$$?

Accepted Answer

$$\frac{dx}{d	au} = \frac{dx^\mu}{d	au} e_\mu$$

Question 26

What is the relation between the totally antisymmetric tensor $$\gamma_{\mu
uho\sigma}$$ and $$\$$gamma_5$$?

Accepted Answer

$$\gamma_{\mu
uho\sigma} = \$$gamma_5$$ \epsilon_{\mu
uho\sigma}$$

Question 27

What is the Clifford algebra form of the inhomogeneous Maxwell equation for the field $$F$$ and current $$j$$?

Accepted Answer

$$\$$partial^2$$ F = \partial \wedge j$$

Question 28

What is the equation of motion for the polyvector $$P in $$the presence of a field $$F$$?

Accepted Answer

$$\dot{P} \equiv \frac{dP}{d	au} = \frac{e}{2m} [P, F]$$

Question 29

What is the Hamiltonian for the unconstrained theory with $$L = (1/2\Lambda)(p^\mu p_\mu - \$$kappa^2$$)$$?

Accepted Answer

$$H = \frac{\Lambda}{2}(p^\mu p_\mu - \$$kappa^2$$)$$

Question 30

What is the Schrödinger equation for the wave function $$\psi in $$the unconstrained theory?

Accepted Answer

$$i \frac{\partial \psi}{\partial 	au} = \frac{\Lambda}{2}(p^\mu p_\mu - \$$kappa^2$$) \psi$$

Question 31

What is the Schrödinger equation for the wave function $$\psi in $$the square-root Hamiltonian theory?

Accepted Answer

$$i \frac{\partial \psi}{\partial s} = \sqrt{p^\mu p_\mu - \$$kappa^2$$} \psi$$

Question 32

How does the Lagrangian $$L$$ transform under a gauge transformation $$\phi$$?

Accepted Answer

$$L 	o L' = L + \frac{d\phi}{d	au}$$

Question 33

How does the scalar gauge field $$V$$ transform under a gauge transformation $$\phi in $$the $$s-$$parametrized theory?

Accepted Answer

$$eV' = eV + \frac{\partial \phi}{\partial s}$$

Question 34

What is the condition for the background fields $$A_\mu$$ and $$V to $$ensure conservation of kinetic momentum?

Accepted Answer

$$\frac{\partial A_\mu}{\partial s} = 0$$, $$\partial_\mu V = 0$$

Question 35

What is the matrix representation of the basis spinors $$\Phi_$$+^{ab}$$ and $$\Phi_$$-^{ab}$$?

Accepted Answer

$$\Phi_$$+^{ab}$$ = \begin{pmatrix} \Phi_+^1$ & 0 \ \Phi_+^2$ & 0 \end{pmatrix}$$, $$\Phi_$$-^{ab}$$ = \begin{pmatrix} 0 & \Phi_-^1$ \ 0 & \Phi_-^2$ \end{pmatrix}$$

Question 36

What is the matrix representation of the Pauli matrix $$\$$sigma_1$$?

Accepted Answer

$$(\$$sigma_1$$)^{ab}$$ = \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix}$

Question 37

How does a spinor $$\psi$$ transform under a rotation $$R$$?

Accepted Answer

$$\psi' = R\psi 	o \begin{pmatrix} \psi'^1 & 0 \ \psi'^2 & 0 \end{pmatrix}$$

Question 38

What is the action of $$e^0$$ and $$e^1 on $$the basis spinor $$u_0$$?

Accepted Answer

$$e^0$$ $$u_0$$ = -$$u_0$$, $$e^1$$ $$u_0 = i$$ $$u_3$$

Question 39

What is the canonical momentum $$\pi(s, x)$$ conjugate to $$\phi in $$the Stueckelberg field theory?

Accepted Answer

$$\pi(s, x) = \frac{\partial \mathcal{L}}{\partial \partial \phi / \partial s} = -\frac{\partial \phi}{\partial s}$$

Question 40

What is the canonical momentum $$\pi_j$$ conjugate to $$\phi_j(x) in $$the Stueckelberg field theory?

Accepted Answer

$$\pi_j = -i \frac{\delta}{\delta \phi_j(x)}$$

Question 41

What is the functional Schrödinger equation for the wave functional $$\Psi in $$the Stueckelberg field theory?

Accepted Answer

$$i \frac{\partial \Psi}{\partial s} = H \Psi$$

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