# 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 $\psi $, so leading to the notion of entanglement of
states and the indeterministic reduction of the wave packet. Once
$\psi $ is determined it is possible to compute the probability of
measurable outcomes, at the same time modifying $\psi $ 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 $dimH\ge 3$, 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 |

Related topic | QuantumChromodynamicsQCD |

Related topic | QuantumOperatorAlgebrasInQuantumFieldTheories |

Related topic | QFTOrQuantumFieldTheories |

Related topic | QuantumSpaceTimes |

Related topic | RichardFeynman |

Related topic | Quantization |

Related topic | FoundationsOfQuantumFieldTheories |

Related topic | FoundationsOfQuantumFieldTheories |

Related topic | QuantumChromodynamicsQCD |

Defines | electromagnetic interactions theory |