(Quantum) reference frames, relational observables, gauge reduction and physical interpretation

Abstract

It is mandatory to know how to operationally define and translate a reference frame into mathematics, in order that a physical interpretation of theory calculations in terms of observational data is possible. The situation is particularly challenging for gauge systems such as General Relativity where spacetime coordinates are subject to spacetime diffeomorphisms considered as gauge transformations turning coordinates into non-observables. This motivates the idea of operationally defined (material) reference frames which specify coordinates in terms of matter or geometry reference fields leading to the concept of relational observables, relational reference frames and gauge reduction. Upon quantisation, all fields become operator valued distributions. Now new conceptual and technical questions arise such as: Should one reduce before or after quantisation and how are the reference fields quantised respectively in either route? Is a reference frame itself subject to quantisation and how are different quantum reference frames related? How does the gauge reduction fit into this, i.e. how can it be that a certain reference field is considered a non-observable in one reference frame and an observable in another which upon quantisation even displays fluctuations? How precisely are gauge dependent fields interpreted in terms of the relational observables in a given reference frame? What is the relative dynamics, e.g. how exactly are physical Hamiltonians of two relational reference frames related? The present conceptual work addresses these and related questions in a non-perturbative field theory context of sufficient generality to cover General Relativity coupled to standard matter. A central role is played by the concept of the relational reference frame transformation (RRFT) for which a general formula is derived and its properties are explored.

Figures (1)

• Figure 1: The branch of the constraint surface $\bar{\Gamma}_s$ defined by $y+h(x,p,q)=0\;\Leftrightarrow \; p+\hat{h}(q,x,y)$ of a 4-dimensional phase space $\Gamma_s$. The $y$ direction is suppressed. It is filled by the congruence of gauge orbits $\gamma\in \hat{\Gamma}_s$ (blue) of the abstract reduced phase space. Displayed are two gauge cuts $\hat{\Gamma}_{s,t},\;\hat{\Gamma}_{s,t'}$ (green) corresponding to the reference frame $(x,k(.))$ and one gauge cut $\widehat{\hat{\Gamma}}_{s,\hat{t}}$ (red) corresponding to the reference frame $(q,\hat{k}(.))$. All gauge orbits lie transversal to all gauge cuts of both foliations $t\mapsto \hat{\Gamma}_{s,t},\; \hat{t}\mapsto \widehat{\hat{\Gamma}}_{s,\hat{t}}$ of $\bar{\Gamma}_s$. The two first gauge cuts are coordinatised by true degrees of freedom $(q,p)$, the latter is coordinatised by true degrees of freedom $(x,y=-h(x,\hat{k}(.),p)$. The relational reference frame transformation $\hat{z}=S_{t,\hat{t}}(z)$ is obtained by picking a point $z\in \hat{\Gamma}_{s,t}$, determining the gauge orbit $\gamma$ on which it lies and determining its intersection point $\hat{z}=\gamma\cap \widehat{\hat{\Gamma}}_{s,\hat{t}}$. The trajectory of $z\mapsto z'=\alpha_{t,t'}(z)$ (magenta) encoded by the $(x,k(.))$ reference frame is by determining the intersection point $z'=\gamma\cap \Gamma_{s,t'}$ of the same gauge orbit $\gamma$ with a leaf of the same foliation at a later time $t'$ and projecting the segment of $\gamma$ between $z,z'$ into the $x=0$ plane.

Theorems & Definitions (1)

• Definition 3.1