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 levelsbeyond ${10}^{-29}$, and thus it is one of the most precise, if not the most precise, physical theoriesthat however does not encompass gravity.

1.1 Measurements and Quantum Field Theories

The question of measurement in quantum mechanics (QM) andquantum field theory (QFT) (http://planetmath.org/QFTOrQuantumFieldTheories)has flourished for about 75 years. The intellectual stakes have been dramatically high, and the problemrattled the development of 20th (and 21st) century physics at thefoundations. Up to 1955, Bohr’s Copenhagen school dominated theterms and practice of quantum mechanics having reached (partially)eye–to–eye with Heisenberg on empirical grounds, although not thecase with Einstein who was firmly opposed on grounds onincompleteness with respect to physical reality. Even to thepresent day, the hard philosophy of this school is respectedthroughout most of theoretical physics. On the other hand, post1955, the measurement problem adopted a new lease of life when vonNeumann’s beautifully formulated QM in the mathematically rigorouscontext of Hilbert spaces. Measurement it was argued involved theinfluence of the Schrödinger equation for time evolution of thewave function $\psi$, so leading to the notion of entanglement ofstates and the indeterministic reduction of the wave packet. Once$\psi$ is determined it is possible to compute the probability ofmeasurable outcomes, at the same time modifying $\psi$ relative tothe probabilities of outcomes and observations eventually causesits collapse. The well–known paradox of Schrödinger’s cat andthe Einstein–Podolsky–Rosen (EPR) experiment are questionsmooted once dependence on reduction of the wave packet isjettisoned, but then other interesting paradoxes have shown theirfaces. Consequently, QM opened the door to other interpretationssuch as ‘the hidden variables’ and the Everett–Wheeler assignedmeasurement within different worlds, theories not without theirrespective shortcomings.

Arm–in–arm with the measurement problem goes a problem of‘the right logic’, for quantum mechanical/complex biological systemsand quantum gravity. It is well–known that classical Booleantruth–valued logics are patently inadequate for quantum theory.Logical theories founded on projections and self–adjointoperators 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 Booleansystem of ‘truth values’. In Butterfield and Isham (1999)–(2004)self–adjoint operators on $H$ with purely discrete spectrum areconsidered. The KS theorem is then interpreted as saying that aparticular presheaf does not admit a global section. Partialvaluations corresponding to local sections of this presheaf areintroduced, and then generalized evaluations are defined. Thelatter enjoy the structure of a Heyting algebra and so comprise anintuitionistic logic. Truth values are describable in terms ofsieve–valued maps, and the generalized evaluations are identifiedas subobjects in a topos. The further relationship with intervalvaluations motivates associating to the presheaf a von Neumannalgebra where the supports of states on the algebra determinesthis relationship.

We turn now to another facet of quantum measurement. Note firstthat QFT pure states resist description in terms of fieldconfigurations since the former are not always physicallyinterpretable. Algebraic quantum field theory (AQFT) as expoundedby Roberts (2004) points to various questions raised byconsidering theories of (unbounded) operator –valueddistributions and nets of von Neumann algebras. Using in part agauge theoretic approach, the idea is to regard two field theoriesas equivalent when their associated nets of observables areisomorphic. More specifically, AQFT considers taking (additive)nets of field algebras over subsets of Minkowski space, which amongother properties, enjoy Bose–Fermi commutation relations. Althoughat first glances there may be analogs with sheaf theory, thesesanalogs are severely limited. The typical net does not give rise toa presheaf because the relevant morphisms are in reverse. Closerthen is to regard a net as a precosheaf, but then the additivitydoes not allow proceeding to a cosheaf structure. This may reflectupon some incompatibility of AQFT with those aspects of quantumgravity (QG) where for example sheaf–theoretic/topos approachesare advocated (as in e.g. Butterfield and Isham (1999)–(2004)).