2.1.1 p=0, D=3, Non-Relativistic Point-Particle

$$p=0, \quad V_p=0,\quad \sigma^{ \mu_1\dots \mu_{p+1}}\equiv 0 \longrightarrow \hbox{pointlike particle}$$ (2)

$$D=3 \longrightarrow \hbox{euclidean $3$-space}$$ (3)

$$m_{p+1} V_p \longrightarrow M_0 = \hbox{particle's mass}$$ (4)

$$\Omega_1=\int_0^T d\tau e(\tau) \hbox{proper time interval}$$ (5)

$$e( \tau )=1\rightarrow \Omega_1= T \longrightarrow \mathrm{\lq\lq gauge fixed''\ proper\ time\ interval}$$ (6)

$$(\sim \mathrm{non-relativistic\ time\ interval})$$

$$Z_{\phi , A}\equiv 1, \hbox{Quenching}$$ (7)

The triviality of $Z^{\mathrm{BULK}}$ follows from the classical equations of motion

$${d P_i\over dt }=0$$ (8)

and the entire dynamics of the system is governed by the zero mode alone

$$P_i(t)= {\mathrm{const.}} \equiv q_i\ .$$ (9)

No other modes contribute to the bulk path-integral which is simply the unit constant. Accordingly, the propagation kernel from $\vec x_0$ to $\vec x$, in a lapse of time $T$, can be computed as proposed in [16] and turns out to have the form

$$K_0\left[\, \vec x - \vec x_0\ ; T \, \right]=\left[\, {M_0\... ...vec x -\vec x_0 \, \right)^{2}-i{M_0\over 2 }\, T\, \right\} ,$$ (10)

from which we obtain the corresponding, non-relativistic, propagator

$$G\left[\, \vec x -\vec x_0\ ; E \, \right] = \int_0^\infty dT\exp\left\{\, i\, T\, E\, \right\}\, \left[ \, {M... ...t\{\, {i M_0\over 2 T}\, \left(\, \vec x - \vec x_0\, \right)^{ 2} \, \right\},$$ (11)

$$E \equiv \Lambda - {M_0\over 2} .$$ (12)