Robert SpekkensDepartment of Applied Mathematics and Theoretical

Physics University of Cambridge

Quantum Mechanics over the real field

July 24, 2007 FQXi

Funding by:The Royal Society

Questions about the utility of Real QM:

Might there be experimental situations where the statistics are well described by Real QM?

Might it be that *all* experimental statistics can be explained using Real QM, and that the use of Real vs. Complex is a conventional choice of the physicist?

Questions about the utility of QM with a superselection rule:

Might there be experimental situations where the statistics are well described by QM with a SSR?

Might it be that *all* experimental statistics can be explained using QM with a SSR, and that the use of QM with vs. without SSR is a conventional choice of the physicist?

YES

YES

QM with a superselection rule

Standard view: some conserved quantities are axiomatically superselected, others not

No SSR SSR

Linear momentumAngular momentumPhoton number

ChargeAtom numberunivalence

no RF

QM with SSR is appropriate if one lacks a reference frame

2SO(3) Unknown!

QM with SSR can describe *any* experiment

External RF paradigmStatesMeasurements

Internal RF paradigmStatesMeasurements

and group invariant

One can find a mapping such that

Define

with

where

and

See: Kitaev, Mayers and Preskill, Phys. Rev. 69, 052326 (2004) Bartlett, Rudolph and Spekkens, Rev. Mod. Phys. 79, 555 (2007)

QM with SSR can always be used if one wishes

External RF paradigmStatesMeasurements

Internal RF paradigmStatesMeasurements

and group invariant

Whether internal RF paradigm or external RF paradigm is used is a conventional choice of the physicist

See: Aharonov and Susskind, Phys. Rev. 155, 1428 (1967)Bartlett, Rudolph and Spekkens, Int. J. Quantum Inf. 4, 17 (2006).

Real Quantum Mechanics

Without resource

Can Real QM emerge by lacking some resource?

Some resource

Relative arrow of time unknown?

Can Real QM describe *any* experiment?

External resource paradigmStatesMeasurements

Internal resource paradigmStatesMeasurements

and real

Can one find a mapping such that

Some resource

Some resource

Define

where

External resource paradigmStatesMeasurements

Internal resource paradigmStatesMeasurements

and real

Some resource

Some resource

If so, then whether external or internal paradigm is used is a conventional choice of the physicisti.e. no matter of fact about nature of field!

Can Real QM describe *any* experiment?