Semi-classical Locality for the Non-relativistic Path Integral in Configuration Space

Abstract

In an accompanying paper Gomes (arXiv:1504.02818, 2015), we have put forward an interpretation of quantum mechanics based on a non-relativistic, Lagrangian 3+1 formalism of a closed Universe M, existing on timeless configuration space $\mathcal {Q}$ of some field over M. However, not much was said there about the role of locality, which was not assumed. This paper is an attempt to fill that gap. Locality in full can only emerge dynamically, and is not postulated. This new understanding of locality is based solely on the properties of extremal paths in configuration space. I do not demand locality from the start, as it is usually done, but showed conditions under which certain systems exhibit it spontaneously. In this way we recover semi-classical local behavior when regions dynamically decouple from each other, a notion more appropriate for extension into quantum mechanics. The dynamics of a sub-region O within the closed manifold M is independent of its complement, $M-O$, if the projection of extremal curves on $\mathcal {Q}$ onto the space of extremal curves intrinsic to O is a surjective map. This roughly corresponds to $e^{i\hat{H}t}\circ \mathsf {pr}_{\mathrm{O}}= \mathsf {pr}_{\mathrm{O}}\circ e^{i\hat{H}t}$, where $\mathsf {pr}_{\mathrm{O}}:\mathcal {Q}\rightarrow \mathcal {Q}_O^{\partial O}$ is a linear projection. This criterion for locality can be made approximate—an impossible feat had it been already postulated—and it can be applied for theories which do not have hyperbolic equations of motion, and/or no fixed causal structure. When two regions are mutually independent according to the criterion proposed here, the semi-classical path integral kernel factorizes, showing cluster decomposition which is the ultimate aim of a definition of locality.

Appendix

1.1 A.1 Principal Fiber Bundles

Principal fiber bundles are useful in the discussion of symmetry groups.

A principal fiber bundle is a manifold P, on which a Lie group G acts freely: $P\times G\rightarrow P$, here we will assume it acts on the left, $(p, g)\rightarrow g\cdot p=L_g(p)$. The space $\{g\cdot p\,~|~g\in G\}$ is called the the fiber (through p). Identifying $p\sim g\cdot p$, we have a projection operator onto the quotient space $\pi : P\rightarrow P/G$. We will use the square bracket notation to denote the orbit of p, i.e. $[p]=\pi ^{-1}(\pi (p))\subset P$, but since there is a one to one correspondence between orbits and points in P / G, we sometimes abuse notation and write $[p]=\pi (p)\in P/G$.

For some open region $U\in P/G$, with the use of a section $\chi :U\rightarrow P$, we can trivialize the bundle: $\pi ^{-1}(U)\simeq \chi (U)\times G$, which allows us to write $P\ni p= (x\,,\, g)$ for $x\in U$. Sections intersect each orbit of the gauge group only once. Given two sections, they are always uniquely related by a function $\lambda :U\rightarrow G$, as in $\chi _1(x)= \lambda (x)\cdot \chi _2(x)$.

Given $\mathfrak {v}\in \mathfrak {g}$, where $\mathfrak {g}:=T_{ \text{ Id }}G$, we define the fundamental vector field associated to it as $T_pP\ni \mathfrak {v}_p^\#:=\frac{d}{dt}_{|t=0}\exp {(t\mathfrak {v})}\cdot p$. The vertical subspace $V_p\subset T_pP$ is the tangent space to the fiber at p, i.e. $V_p=\text{ span }\{\mathfrak {v}_p^\#\,~|~\mathfrak {v}\in \mathfrak {g}\}$.

In standard gauge field theory, the space of physically equivalent field configurations requires that one quotient the total space of field configurations by the total group of gauge transformations, $\mathcal {A}/$Gau(G), where $\mathcal {A}$ is the field space of connections on space-time, G is the internal symmetry group and Gau(G) is the group of space-time dependent gauge-transformations. A choice of gauge is a smooth embedding

$$\begin{aligned} \chi : \mathcal {A}/\text{ Gau }(G)\rightarrow \mathcal {A} \end{aligned}$$

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such that each orbit only intersects the image of $\chi$ once, and it does so transversally, i.e. $\text{ Im }(T_A\chi )\oplus V_A=T_A\mathcal {A}$.

Usually, due for instance to Gribov ambiguities, there is no global section $\chi$, and we need to restrict the section to an open set,

$$\begin{aligned}\chi : \pi (\mathcal {U})\rightarrow \pi ^{-1}(\pi (\mathcal {U}))\end{aligned}$$

where $\pi :\mathcal {A}\rightarrow \mathcal {A}/\text{ Gau }(G)$ is the projection operator onto the quotient space and $\mathcal {U}$ is some open region in $\mathcal {A}$.

1.2 A.2 Slice Theorem for Riem

Ebin and Palais have shown that $\mathcal {Q}$ has a local slice [33, 34]. They did this through the use of the normal exponential map to the orbit along a given point. That is, for an arbitrary given metric $\bar{g}_{ab}$, the orbits were shown to be embedded manifolds, and the normal exponential (according to the supermetric (3) with $\lambda =0$): $\text{ Exp }_{\bar{g}}:W\subset V_{\bar{g}}^\perp \rightarrow \mathcal {Q}$ was shown to be a local diffeomorphism onto its image for a given open set W, where, as can easily be seen from the form of $V_g$ and (3),

$$\begin{aligned} V_{\bar{g}}^\perp =\{u^{ab}\in T_g\mathcal {Q}~|~\bar{\nabla }_a u^{ab}=0\} \end{aligned}$$

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By then showing that the tubular bundle around this orbit was locally diffeomorphic to $\mathcal {Q}$, one has, for $g_{ab}\in \pi ^{-1}(\pi (\text{ Im }(\text{ Exp }_{\bar{g}}(W)))$, a unique $f_g\in \text{ Diff }(M)$ such that

$$\begin{aligned} f^*_g g_{ab} =\text{ Exp }_{\bar{g}}(w_g) \end{aligned}$$

for a unique $w_g\in W$,

$$\begin{aligned} w_g=\text{ Exp }_{\bar{g}}^{-1}(f_g^*g_{ab}) \end{aligned}$$

Thus

$$\begin{aligned} g_{ab}=f^*_g(\text{ Exp }_{\bar{g}}(w_g)) \end{aligned}$$

Furthermore, for $g^2_{ab}=h^*g^1_{ab}$, we then have $f_{g^2}=h^{-1}\circ f_{g^1}$. Thus $w_g=w_{[g]}$ and for any $\tilde{g}_{ab}\in [g_{ab}]\subset \pi ^{-1}\pi (\mathcal {U})$, the section is given by

$$\begin{aligned} \chi (\pi (\tilde{g}_{ab}))= \text{ Exp }_{\bar{g}}(w_{[\tilde{g}]}) \end{aligned}$$

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In more heuristic terms, $\chi$ takes any metric along the orbits and translates it along the orbit until it hits the orthogonal exponential section at the height of $\bar{g}_{ab}$. This intersection gives us the value of $\chi$ for the given equivalence class.

Another way of saying it would be to use the ‘best-matching’ ideas (see e.g. [23]). In this context, the choice of representative of a metric would be the one that requires the ‘least’ deformation to get to your metric (where ‘least’ is wrt some supermetric in Riem).

1.3 A.3 The Jacobi Metric

The Jacobi version of the Maupertuis principle establishes some instances when dynamics can be viewed as geodesic motion in an associated Riemannian manifold. That is, if the action can be written as

$$\begin{aligned} S = \int (T - V)dt\end{aligned}$$

for a system defined in a Riemannian manifold, (M, g), with smooth potential U, the extremal trajectories of S with energy $E = T + U$ coincide with the extremals (geodesics) of the length functional

$$\begin{aligned}L=\int ds \end{aligned}$$

defined in (M, h), where h is the Jacobi metric, conformally related to g by $h = 2(E - V)g$. This I will call the Jacobi procedure, whereby one builds a metric in configuration space whose geodesics are extremal paths of the action.¹⁸

1.4 A.4 The Semi-classical Approximation

Given an action S in the configuration space of some field $\phi$ over a spatial manifold M, one can build the path integral propagator between an initial and final configuration, $\phi _1$ and $\phi _2$ as:

$$\begin{aligned} K[\phi _1,\phi _2]= \int _{\gamma \in \Gamma (\phi _1,\phi _2)} \mathcal {D}\gamma \exp {[iS[\gamma (\lambda )]/\hbar ]} \end{aligned}$$

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where $\Gamma (\phi _1,\phi _2)$ is taken to be the space of paths (in some differentiability class) between $\phi _1$ and $\phi _2$, and I used square brackets to denote functional dependence.

The following semi-classical, or saddle point, approximation for the path integral in configuration space was the fundamental object used in [3]. For (locally) extremal paths $\gamma _{\text{ cl }}$ between an initial and a final field configuration, denoting the on-shell action for these paths as $S_{\gamma _{\text{ cl }}}$, accurate for $1<< S_{\gamma _{\text{ cl }}}/\hbar$, it can be written as:

$$\begin{aligned} K_{{\text{ cl }}}[{\phi ^*}, \phi _f]= A \sum _{\gamma _{\text{ cl }}}(\Delta _{\gamma _{\text{ cl }}})^{1/2}\exp {\left( i S_{\gamma _{\text{ cl }}}[{\phi ^*},\phi _f]/\hbar \right) } \end{aligned}$$

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where A is a normalization factor (independent of the initial and final configurations),¹⁹ I used square brackets to denote functional dependence of the on-shell action, and the Van Vleck determinant is now defined as

$$\begin{aligned} \Delta _{\gamma _{\text{ cl }}}:=\det \left( -\frac{\delta ^2 S_{\gamma _{\text{ cl }}}[{\phi ^*},\phi _f]}{\delta {\phi ^*}(x)\delta \phi _f(y)}\right) = \det \left( \frac{\delta \pi ^\gamma _f[{\phi ^*}; x)}{\delta {\phi ^*}(y)}\right) \end{aligned}$$

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where we used DeWitt’s mixed functional/local dependence notation $[{\phi ^*}; x)$, and the on-shell momenta is defined as

$$\begin{aligned}\pi ^\gamma _f[{\phi ^*}; x):= -\frac{\delta S_{\gamma _{\text{ cl }}}[{\phi ^*},\phi _f]}{\delta \phi _f(x)} \end{aligned}$$

The Van-Vleck matrix is degenerate when gauge-symmetries exist, in which case one must provide a suitable gauge-fixing. If it is not degenerate, it embodies the spread of classical trajectories.