quantum electrodynamics

quantum electrodynamics

1

Quantum electrodynamics (

Q.E.D)

Q.E.D (

http://planetmath.org/QEDInTheoreticalAndMathematicalPhysics

) is the advanced, standard mathematical and quantum physics treatment of electromagnetic interactions through several approaches, the more advanced including the path-integral approach by Feynman, Dirac’s

Operator

and QED (

http://planetmath.org/QEDInTheoreticalAndMathematicalPhysics

) Equations, thus including either Special or General Relativity formulations of electromagnetic phenomena. More recent approaches have involved spinor (Cartan and Weyl) and twistor (Penrose) representations of Quantum

Hilbert spaces

of

quantum states

and observable quantum oprators. QED results are currently at precision levels
beyond

10

-

29

, and thus it is one of the most precise, if not the most precise, physical theories
that however does not encompass gravity.

1.1

Measurements and Quantum Field Theories

The question of measurement in quantum mechanics (QM) and

quantum field theory

(QFT) (

http://planetmath.org/QFTOrQuantumFieldTheories

)
has flourished for about 75 years. The intellectual stakes have been dramatically high, and the problem
rattled the development of 20th (and 21st) century physics at the

foundations

. Up to 1955, Bohr’s Copenhagen school dominated the
terms and practice of quantum mechanics having reached (partially)
eye–to–eye with Heisenberg on empirical grounds, although not the
case with Einstein who was firmly opposed on grounds on

incompleteness

with respect to physical reality. Even to the
present day, the hard philosophy of this school is respected
throughout most of theoretical physics. On the other hand, post
1955, the measurement problem adopted a new lease of life when von
Neumann’s beautifully formulated QM in the mathematically

rigorous

context

of Hilbert spaces. Measurement it was argued involved the
influence of the Schrödinger equation for time evolution of the

wave function

ψ

, so leading to the notion of entanglement of
states and the indeterministic

reduction

of the wave packet. Once

ψ

is determined it is possible to compute the probability of
measurable outcomes, at the same time modifying

ψ

relative to
the probabilities of outcomes and observations eventually causes
its collapse. The well–known

paradox

of Schrödinger’s cat and
the Einstein–Podolsky–Rosen (EPR) experiment are questions
mooted once dependence on reduction of the wave packet is
jettisoned, but then other interesting paradoxes have shown their
faces. Consequently, QM opened the door to other

interpretations

such as ‘the hidden variables’ and the Everett–Wheeler assigned
measurement within different worlds, theories not without their
respective shortcomings.

Arm–in–arm with the measurement problem goes a problem of
‘the right logic’, for quantum mechanical/complex biological systems
and

quantum gravity

. It is well–known that classical

Boolean

truth–valued logics are patently inadequate for

quantum theory

.
Logical theories founded on projections and self–adjoint
operators on Hilbert space

H

do run in to certain problems . One
‘no–go’

theorem

is that of Kochen–Specker (KS) which for

dim

⁡

H

≥

3

, does not permit an evaluation (global) on a Boolean
system of ‘truth values’. In Butterfield and Isham (1999)–(2004)
self–adjoint operators on

H

with purely discrete spectrum are
considered. The KS theorem is then interpreted as saying that a
particular

presheaf

does not admit a

global section

. Partial
valuations corresponding to local sections of this presheaf are
introduced, and then generalized evaluations are defined. The
latter enjoy the

structure

of a

Heyting algebra

and so comprise an

intuitionistic logic

. Truth values are describable in terms of
sieve–valued maps, and the generalized evaluations are identified
as subobjects in a topos. The further relationship with interval
valuations motivates associating to the presheaf a

von Neumann
algebra

where the

supports

of states on the algebra determines
this relationship.

We turn now to another facet of quantum measurement. Note first
that QFT

pure states

resist description in terms of field

configurations

since the former are not always physically
interpretable. Algebraic quantum field theory (AQFT) as expounded
by Roberts (2004) points to various questions raised by
considering theories of (

unbounded

) operator –valued

distributions

and nets of von Neumann algebras. Using in part a
gauge theoretic approach, the idea is to regard two field theories
as

equivalent

when their associated nets of observables are
isomorphic. More specifically, AQFT considers taking (

additive

)
nets of field algebras over subsets of

Minkowski space

, which among
other properties, enjoy Bose–Fermi commutation

relations

. Although
at first glances there may be analogs with sheaf theory, theses
analogs are severely limited. The typical net does not give rise to
a presheaf because the relevant

morphisms

are in reverse. Closer
then is to regard a net as a precosheaf, but then the additivity
does not allow proceeding to a cosheaf structure. This may reflect
upon some incompatibility of AQFT with those aspects of quantum
gravity (QG) where for example sheaf–theoretic/topos approaches
are advocated (as in e.g. Butterfield and Isham (1999)–(2004)).

Title

quantum electrodynamics

Canonical name

QuantumElectrodynamics

Date of creation

2013-03-22 18:10:30

Last modified on

2013-03-22 18:10:30

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

28

Author

bci1 (20947)

Entry type

Topic

Classification

msc 81Q60

Classification

msc 81Q50

Classification

msc 55U99

Classification

msc 81Q30

Synonym

Q.E.D

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Defines

electromagnetic interactions theory

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