Quantum Mechanics as a Theory of Incompatible Symmetries
- URL: http://arxiv.org/abs/2206.00008v1
- Date: Tue, 31 May 2022 16:04:59 GMT
- Title: Quantum Mechanics as a Theory of Incompatible Symmetries
- Authors: Roger A. Hegstrom and Alexandra J. MacDermott
- Abstract summary: We show how classical probability theory can be extended to include any system with incompatible variables.
We show that any probabilistic system (classical or quantal) that possesses incompatible variables will show not only uncertainty, but also interference in its probability patterns.
- Score: 77.34726150561087
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: It is increasingly becoming realized that incompatible variables, which play
an essential role in quantum mechanics (QM), are not in fact unique to QM. Here
we add a new example, the "Arrow" system, to the growing list of classical
systems that possess incompatible variables. We show how classical probability
theory can be extended to include any system with incompatible variables in a
general incompatible variables (GIV) theory. We then show how the QM theory of
elementary systems emerges naturally from the GIV framework when the
fundamental variables are taken to be the symmetries of the states of the
system. This result follows primarily because in QM the symmetries of the
Poincare group play a double role, not only as the operators which transform
the states under symmetry transformations but also as the fundamental variables
of the system. The incompatibility of the QM variables is then seen to be just
the incompatibility of the corresponding space-time symmetries. We also arrive
at a clearer understanding of the Born Rule: although not primarily derived
from symmetry - rather it is simply a free Pythagorean construction for
accommodating basic features of classical probability theory in Hilbert spaces
- it is Poincare symmetry that allows the Born Rule to take on its familiar
form in QM, in agreement with Gleason's theorem. Finally, we show that any
probabilistic system (classical or quantal) that possesses incompatible
variables will show not only uncertainty, but also interference in its
probability patterns. Thus the GIV framework provides the basis for a broader
perspective from which to view QM: quantal systems are a subset of the set of
all systems possessing incompatible variables (and hence showing uncertainty
and interference), namely the subset in which the incompatible variables are
incompatible symmetries.
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