Symmetry, Reference Frames and Relational Quantities in Quantum Mechanics

Symmetry, Reference Frames and Relational Quantities in Quantum Mechanics Abstract. We propose that observables in quantum theory are properly understood as representatives
of symmetry-invariant quantities relating one system to another, the latter to be called
a reference system. We provide a rigorous mathematical language to introduce and study quantum
reference systems, showing that the orthodox “absolute” quantities are good representatives
of observable relative quantities if the reference state is suitably localised. We use this relational
formalism to critique the literature on the relationship between reference frames and superselection
rules, settling a long-standing debate on the subject.
In classical physics, symmetry, reference frames and the relativity of physical quantities are
intimately connected. The position of a material object is defined as relative to a given frame, and
the relative position of object to frame is a shift-invariant quantity. Galiliean directions/angles,
velocities and time of events are all relative, and invariant only once the frame-dependence has
been accounted for. The relativity of these quantities is encoded in the Galilei group, and the
observable quantities are those which are invariant under its action. Einstein’s theory engendered a
deeper relativity—the length of material bodies and time between spatially separated events are also
frame-dependent quantities—and observables must be sought in accordance with their invariance
under the action of the Poincar´e group.
In quantum mechanics the analogues of those quantities mentioned above (e.g., position, angle)
must also be understood as being relative to a reference frame. As in the normal presentation of the
classical theory, the reference frame-dependence is implicit. However, in the quantum case, there
arises an ambiguity regarding the definition of a reference frame: if it is classical, this raises the
spectre of the lack of universality of quantum mechanics along with technical difficulties surrounding
hybrid classical-quantum systems; if quantum, such a frame is subject to difficulties of definition
and interpretation arising from indeterminacy, incompatibility, entanglement, and other quantum
properties (see, e.g., [1, 2, 3] for early discussions of some of the important issues).
In previous work [4, 5], following classical intuition we have posited that observable quantum
quantities are invariant under relevant symmetry transformations, and examined the properties of
quantum reference frames (viewed as physical systems) which allow for the usual description, in
which the reference frame is implicit, to be recovered. We constructed a map U which brings out
the relative nature of quantities normally presented in “absolute” form in conventional treatments,
which allows for a detailed study of the relativity of states and observables in quantum mechanics
and the crucial role played by reference localisation. The objectives for this paper are: 1) To provide a mathematically rigorous and conceptually clear
framework with which to discuss quantum reference frames, making precise existing work on the
subject (e.g., [6]) and providing proofs of the main claims in [5]; 2) to construct examples, showing
how symmetry dictates that the usual text book formulation of quantum theory describes the
relation between a quantum system and an appropriately localised reference system; 3) to provide
further conceptual context for the quantitative trade-off relations proven in [4]; 4) to provide explicit
and clear explanation of what it means for states/observables to be defined relative to an external
reference frame, and show how such an external description is compatible with quantum mechanics
as a universal theory; 5) to introduce the concepts of absolute coherence and mutual coherence,
showing the latter to be required for good approximation of relative quantities by absolute ones,
and demonstrating it to be the crucial property for interference phenomena to manifest in the
presence of symmetry; 6) to address the questions of dynamics and measurement under symmetry,
offering an interpretation of the Wigner-Araki-Yanase theorem based on relational quantities; 7)
to analyse simplified models similar to those appearing in the literature purporting to produce
superpositions typically thought “forbidden” due to superselection rules, and provide a critical
analysis of large amplitude limits in this context guided by two interpretational principles due
to Earman and Butterfield, leading directly to 8) to provide a historical account of two differing
views on the nature of superselection rules ([7, 8] “versus” [10, 9, 6]), their fundamental status in
quantum theory and precisely what restrictions arise in the presence of such a rule, showing how our
framework brings a unity to the opposing standpoints; 9) to remove ambiguities and inconsistencies
appearing in all previous works on the subject of the connection between superselection rules and
reference frames; 10) to offer a fresh perspective, based on the concept of mutual coherence, on the
nature and reality of quantum optical coherence, settling a long-standing debate on the subject of
whether laser light is “truly” coherent. See also [11] for an important contribution on this topic.
We provide general arguments and many worked examples to show precisely how the framework
presented works in practice, and which simplify a number of models appearing in the literature.
Our paper constitutes further effort in a long line of enquiries (e.g., ([6, 12, 13, 14, 15, 16, 17])
aimed at capturing the relationalism at the heart of the quantum mechanical world view. The
fundamental role of symmetry has not impressed itself strongly upon previous consideration of the
relative nature of the quantum description, and we view this work (along with [4, 5]) as opening
new lines of enquiry in this direction. Our work is inspired by [6] and visits similar themes, and is
complementary to recent work on resource theories (e.g., [6, 18, 19, 20, 21]), which focus primarily
on practical questions surrounding, for example, high-precision quantum metrology.