General Solution and Canonical Quantization of the Conic Path Constrained Second-Class System

Abstract

We consider the problem of constrained motion along a conic path under a given external potential function. The model is described as a second-class system capturing the behavior of a certain class of specific quantum field theories. By exhibiting a suitable integration factor, we obtain the general solution for the associated nonlinear differential equations. We perform the canonical quantization in a consistent way in terms of the corresponding Dirac brackets. We apply the Dirac–Bergmann algorithm to unravel and classify the whole internal constraints structure inherent to its dynamical Hamiltonian description, obtain the proper extended Hamiltonian function, determine the Lagrange multiplier and compute all relevant Poisson brackets among the constraints, Hamiltonian and Lagrange multiplier. The complete Dirac brackets algebra in phase space as well as its physical realization in terms of differential operators is explicitly obtained.

Appendices

Appendix A - Constraint Poisson Brackets

We present here the technical details for obtaining Eqs. (20) and (21) as well as the explicit form for the corresponding functions $\Omega$ and F.

To obtain $\{\chi _3,\lambda \}$ from Eqs. (15) and (14), notice that if we define

$$\begin{aligned} T_x\equiv (2Ax+By+D)\,,~~~T_y\equiv (Bx+2Cy+E)\,, \end{aligned}$$

(80)

$$\begin{aligned} X\equiv T_x (V_{xx}+8Az)+ T_y(V_{xy}+4Bz)+3BV_y+6AV_x \end{aligned}$$

(81)

and

$$\begin{aligned} Y\equiv T_x(V_{xy}+4Bz)+T_y(V_{yy}+8Cz) +6CV_y + 3BV_x \, , \end{aligned}$$

(82)

we may write

$$\begin{aligned} \{\chi _3,\lambda \}= -\frac{1}{m^2M^2}\left[ p_x \Omega ^{x} + p_y \Omega ^{y} \right] \, , \end{aligned}$$

(83)

with

$$\begin{aligned} \Omega ^x &= M(-T_x X_x + 2AX + BY - T_y X_y)\\ &~~~~~~~~~~+X(T_xM_x+T_y M_y)\,, \end{aligned}$$

(84)

and

$$\begin{aligned} \Omega ^y &= M(-T_xY_x+BX-T_yY_y+2CY) \\&~~~~~~~~~~+ Y(T_xM_x+T_yM_y)\,. \end{aligned}$$

(85)

Then, an explicit calculation gives

$$\begin{aligned} \Omega ^x=\; & {} \Big [ 6AT_x^2 + 4B T_xT_y + (2A+4C)T_y^2 \Big ] X \nonumber \\&+ BMY\nonumber -M \Big [ T_x(8AV_{xx}+4BV_{xy}\\&+(16A^2+4B^2)z + T_x V_{xxx} + T_y V_{xxy}) \nonumber \\&+T_y ((2C+6A)V_{xy} + 8B(A+C)z \nonumber \\&+ T_x V_{xxy} + T_y V_{xyy} + BV_{xx} + 3B V_{yy}) \Big ] \end{aligned}$$

(86)

and

$$\begin{aligned} \Omega ^y= \; & \Big [ (4A+2C)T_x^2 + 4B T_xT_y + 6CT_y^2 \Big ] Y \nonumber \\&+ BMX- M \Big [ T_x ((2A+6C)V_{xy} \nonumber \\&+ 8B(A+C)z + T_x V_{xxy} + T_y V_{xyy} \nonumber \\&+ BV_{yy} + 3B V_{xx}) + T_y(8CV_{yy}+4BV_{xy}\nonumber \\&+(16C^2+4B^2)z + T_y V_{yyy} + T_x V_{xyy}) \Big ] \, , \end{aligned}$$

(87)

which corresponds to the expressions for $\Omega ^{x}(x,y,z)$ and $\Omega ^{y}(x,y,z)$ within equation (20).

Concerning $\{\chi _4,\lambda \}$, note that if we define further

$$\begin{aligned} P\equiv Ap_x^2+Bp_xp_y+Cp_y^2\,, \end{aligned}$$

(88)

we may write

$$\begin{aligned} \{\chi _4,\lambda \}=\frac{2}{m^2}\{P,\lambda \}-\frac{z}{m}\{ M,\lambda \} -\frac{1}{m} \{N,\lambda \} \end{aligned}$$

(89)

with

$$\begin{aligned} \frac{2}{m^2}\{P,\lambda \}=p_x^2F_{(2)}^{xx}+p_xp_yF_{(2)}^{xy}+p_y^2F_{(2)}^{yy} \, , \end{aligned}$$

(90)

where

$$\begin{aligned} F_{(2)}^{xx}=\;& \frac{2}{m^3M^2}\Big [ (2AX_x+BX_y)M\\&-2X\Big ( (4A^2+B^2)T_x+2B(A+C)T_y \Big ) \Big ] \, , \end{aligned}$$

(91)

$$\begin{aligned} F_{(2)}^{yy}=\;& \frac{2}{m^3M^2}\Big [ (BY_x+2CY_y)M\\&-2Y\Big ( 2B(A+C)T_x + (4C^2+B^2)T_y \Big ) \Big ] \end{aligned}$$

(92)

and

$$\begin{aligned} F_{(2)}^{xy}= & {} \frac{2}{m^3M^2}\Big [ (2AY_x+BX_x+BY_y+2CX_y)M\\&-2(BX+2AY)(2AT_x+BT_y) \nonumber -2(BY\\&+2CX)(BT_x+2CT_y)\Big ] \end{aligned}$$

(93)

correspond to the functions defined in Eq. (21). Furthermore, note that the remaining momenta-independent part of $\{\chi _4,\lambda \}$ is given by

$$\begin{aligned} F_{(0)}& = -\frac{z}{m}\{ M,\lambda \} -\frac{1}{m} \{N,\lambda \} \\& = \frac{1}{m^2M} \Big [ X(zM_x+N_x)+Y(zM_y+N_y) \Big ] \end{aligned}$$

(94)

corresponding to $F_{(0)}$ as defined in Eq. (21).

Appendix B - DBs for the Auxiliary Variable z

Concerning the auxiliary variable z introduced in Eq. (3) to implement the restriction due to Eq. (1), we have the non-null DBs

$$\begin{aligned} \{x,z\}^*&=2m^{-1}M^{-1}\big (Bx+2Cy+E\big ) \{p_x,p_y\}^*\,,\\ \{y,z\}^*&=2m^{-1}M^{-1}(2Ax+By+D) \{p_y,p_x\}^*\, , \end{aligned}$$

(95)

and

$$\begin{aligned} \{z,p_x\}^*&{}=-2m^{-1} \left( 2Cp_y+Bp_x\right) \{p_x,p_y\}^* \\ &{} \; \, ~~~+M^{-2}(Bx+2Cy+E) \Big [ V_x \bar{K}(y)-V_yK(x) \\ &{} \; \, ~~~ +\big ((2Ax+By+D)^2 -(Bx+2Cy+E)^2\big )\big (V_{xy}+2zB\big ) \\ &{} \; \, ~~~ -(2Ax+By+D)(Bx+2Cy+E)\big (V_{xx}-V_{yy}+2A-2C\big ) \Big ] \, ,\\ \{z,p_y\}^*&{}=2m^{-1} \left( 2Ap_x+Bp_y\right) \{p_x,p_y\}^* \\ &{} \; \, ~~~-M^{-2}(2Ax+By+D) \Big [ V_x \bar{K}(y)-V_yK(x) \\ &{} \; \, ~~~ +\big ((2Ax+By+D)^2 -(Bx+2Cy+E)^2\big )\big (V_{xy}+2zB\big ) \\ &{} \; \, ~~~ -(2Ax+By+D)(Bx+2Cy+E)\big (V_{xx}-V_{yy}+2A-2C\big ) \Big ] \, , \end{aligned}$$

(96)

which also follow from definition (57). As we can see, the derivatives of the potential function V(x, y) show up only in the Dirac brackets among z and the momenta in Eq. (96). Of course, the DB between any phase space function and $p_z$ is null, as the latter constitutes one of constraints (15) in itself.