## Algebra of Classical and Quantum Binary Measurements

**Authors:** C A Brannen

The simplest measurements in physics are binary; that is, they have only two possible
results. An example is a beam splitter. One can take the output
of a beam splitter and use it as the input of another beam splitter. The compound measurement
is described by the product of the Hermitian matrices that describe the beam splitters.
In the classical case the Hermitian matrices commute (are diagonal) and the measurements
can be taken in any order. The general quantum situation was described by Julian Schwinger
with what is now known as ``Schwinger's Measurement Algebra''. We simplify his results
by restriction to binary measurements and extend it to include classical as
well as imperfect and thermal beam splitters. We use elementary methods to introduce
advanced subjects such as geometric phase, Berry-Pancharatnam phase, superselection sectors,
symmetries and applications to the identities of the Standard Model fermions.

**Comments:** 22 Pages. Added derivation of complex numbers in QM and minor changes. As submitted to journal.

**Download:** **PDF**

### Submission history

[v1] 2018-01-25 01:08:02

[v2] 2018-01-30 23:57:24

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