quantum field theories (QFT)

quantum field theories (QFT)

This topic links the general framework of

quantum field theories

to group

symmetries

and other relevant mathematical concepts utilized to

represent

quantum fields
and their fundamental properties.

0.1

Fundamental, mathematical concepts in quantum field theory

Quantum field theory (QFT)

is the general framework for describing the physics of relativistic

quantum systems

, such as, notably, accelerated elementary particles.

Quantum electrodynamics

(QED)

, and QCD or quantum chromodynamics (

http://planetmath.org/QCDorQuantumChromodynamics

) are only two distinct theories among several quantum field theories, as their fundamental representations correspond, respectively, to very different–

U

⁢

(

1

)

and

S

⁢

U

⁢

(

3

)

– group symmetries. This obviates the need for ‘more fundamental’ , or

extended quantum symmetries

, such as those afforded by either larger groups such as

U

⁢

(

1

)

×

S

⁢

U

⁢

(

2

)

×

S

⁢

U

⁢

(

3

)

or spontaneously broken, special symmetries of a less restrictive kind present in ‘

quantum groupoids

’ as for example in

weak Hopf algebra

representations

, or in

locally compact groupoid

,

G

l

⁢

c

unitary representations

, and so on, to the higher dimensional (quantum) symmetries of

quantum double

groupoids

, quantum double algebroids,

quantum categories

,quantum

supercategories

and/or quantum (

supersymmetry

) superalgebras (or graded ‘Lie’

algebras

); see, for example, their full

development

in a recent QFT textbook

[

4

]

that lead to superalgebroids in

quantum gravity

or QCD.

References

1

A. Abragam and B. Bleaney.:

Electron Paramagnetic Resonance of Transition Ions.

Clarendon Press: Oxford, (1970).

2

E. M. Alfsen and F. W. Schultz:

Geometry of

State Spaces

of
Operator Algebras

, Birkhäuser, Boston–Basel–Berlin (2003).

3

D.N. Yetter., TQFT’s from homotopy 2-types.

J. Knot Theor

.

2

: 113–123(1993).

4

S. Weinberg.:

The

Quantum Theory

of Fields

. Cambridge, New York and Madrid:
Cambridge University Press, Vols. 1 to 3, (1995–2000).

5

A. Weinstein : Groupoids: unifying internal and external symmetry,

Notices of the Amer. Math. Soc.

43

(7): 744–752 (1996).

6

J. Wess and J. Bagger:

Supersymmetry and Supergravity

,
Princeton University Press, (1983).

7

J. Westman: Harmonic analysis on groupoids,

Pacific J. Math.

27

: 621-632. (1968).

8

J. Westman: Groupoid theory in algebra,

topology

and analysis.,

University of California at Irvine

(1971).

9

S. Wickramasekara and A. Bohm: Symmetry representations in the

rigged Hilbert space

formulation of quantum mechanics,

J. Phys. A

35

(3): 807-829 (2002).

10

Wightman, A. S., 1956, Quantum Field Theory in Terms of Vacuum Expectation Values, Physical Review,

101

: 860–866.

11

Wightman, A.S. and Garding, L., 1964, Fields as Operator–Valued

Distributions

in Relativistic Quantum Theory, Arkiv für Fysik, 28: 129–184.

12

S. L. Woronowicz : Twisted

SU(2)

group : An example of a non–

commutative

differential calculus, RIMS, Kyoto University

23

(1987), 613–665.

Title

quantum field theories (QFT)

Canonical name

QuantumFieldTheoriesQFT

Date of creation

2013-03-22 18:10:52

Last modified on

2013-03-22 18:10:52

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

35

Author

bci1 (20947)

Entry type

Topic

Classification

msc 55U99

Classification

msc 81T80

Classification

msc 81T75

Classification

msc 81T70

Classification

msc 81T60

Classification

msc 81T40

Classification

msc 81T25

Classification

msc 81T18

Classification

msc 81T13

Classification

msc 81T10

Classification

msc 81T05

Synonym

quantum theories

Related topic

QEDInTheoreticalAndMathematicalPhysics

Related topic

QuantumChromodynamicsQCD

Related topic

Algebroids

Related topic

Distribution4

Related topic

AlgebraicQuantumFieldTheoriesAQFT

Related topic

Quantization

Related topic

QuantumChromodynamicsQCD

Defines

quantum interactions of all kinds

Defines

minus gravitational ones

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