spin networks and spin foams

spin networks and spin foams

Definition 0.1

.

spin networks

are one-dimensional

C

⁢

W

complexes consisting of quantum spin states of particles, defined by elements of Pauli matrices represented as vertices of a directed graph or network, and with the edges of the network representing the

connections

, or links, between such quantum spin states.

Remark 0.1

.

On

current

formal

definitions

of spin networks.

For

quantum systems

with known standard

symmetry

formal definitions of spin networks have also been reported in terms
of

symmetry group

representations

. An example of such a formal definition in terms of

Lie group
representations

on

Hilbert spaces

of quantum states and

operators

is provided next.

Definition 0.2

.

Spin networks

are formally defined here for quantum systems with ‘standard’* quantum symmetry in terms of

Lie group

(

G

L

)

irreducible representations

(

http://planetmath.org/GroupRepresentation

) on complex Hilbert spaces

ℋ

of quantum states and observable operators; such representations are precisely defined by special group

homomorphisms

as follows.
Consider

R

⁢

e

as a Lie group

G

L

, and also consider the complex Hilbert space

ℋ

to be

B

⁢

[

ℋ

]

, the group of

bounded linear operators

of

ℋ

which have a

bounded

inverse

, and more specifically to be

L

2

⁢

(

R

⁢

e

)

.
Then, one defines the

G

L

-representation

as the

group homomorphism

ρ

:

R

⁢

e

→

B

⁢

[

L

2

⁢

(

R

⁢

e

)

]

with

ρ

⁢

(

r

)

:

{

f

⁢

(

x

)

}

↦

f

⁢

(

r

-

1

⁢

x

)

, where

r

∈

R

⁢

e

and

f

⁢

(

x

)

∈

L

2

⁢

(

R

⁢

e

)

.

*The word ‘standard’ is employed here with the meaning of the Standard Model of Physics (SUSY) which does

not

include either

Quantum Gravity

or its

extended quantum symmetries

.

Definition 0.3

.

spin foams

are two-dimensional

C

⁢

W

complexes representing two local
spin networks as described in

Definition 0.1

with quantum transitions between them;

spin foams

are sometimes also represented by

functors

of spin networks considered as (small)

categories

(

viz.

Baez and Dollan,1998a,b;

[

3

,

4

]

).

For the sake of completeness, let us recall here the following

Definition 0.4

.

a

C

⁢

W

complex

,

X

c

is a

topological space

which is the union of an expanding

sequence

of

subspaces

X

n

such that, inductively,

X

0

is a

discrete set

of points called
vertices and

X

n

+

1

is the

pushout

obtained from

X

n

by attaching disks

D

n

+

1

along
“

attaching maps

”

j

:

S

n

→

X

n

. Each resulting map

D

n

+

1

⟶

X

is
called a

cell

. The subspace

X

n

is called the “

n

-

skeleton

” of X.

An Example

of a

C

⁢

W

complex is a graph or ‘network’ regarded as a one-dimensional

C

⁢

W

complex.

Remark:

Such ‘purely’ topological definitions seem to miss much of the associated quantum operator

algebraic structures

that are essential to the

mathematical foundation

of

quantum theories

; note however the first related entry
that addresses this important, algebraic question.

Note.

The

concepts

of

spin networks

and

spin foams

were recently developed in the context
of

Mathematical Physics

as part of the more general effort of attempting to formulate mathematically a concept of

Quantum State Space

which is also applicable, or relates to

Quantum Gravity

spacetimes. The

spin observable

– which is fundamental in quantum theories– has no corresponding concept in classical mechanics. (However, classical

momenta

(both linear and angular) have corresponding quantum observable operators that are quite different in form, with their eigenvalues taking on different sets of values in Quantum Mechanics than the ones might expect from classical mechanics for the ‘corresponding’ classical observables); the spin is an

intrinsic

observable of all massive quantum ‘particles’, such as electrons, protons, neutrons, atoms, as well as of all field quanta, such as photons,

gravitons

, gluons, and so on; furthermore, every quantum ‘particle’ has also associated with it a de Broglie wave, so that it cannot be realized, or ‘pictured’, as any kind of classical ‘body’. This

intrinsic

, spin observable, can also be understood as an

missing

internal symmetry of quantum particles, which in many cases can be understood in terms of ‘internal’ symmetry group representations, such as the Dirac or Pauli matrices that are currently employed in quantum mechanics,

Quantum Electrodynamics

, QCD and QFT. There are thus

fermion

(quantum) symmetries, quantum statistics, etc, for quantum particles with half-integer spin values (for massive particles such as electrons, protons,neutrons, quarks, nuclei with an

odd

number of nucleons) and

boson

(quantum) symmetries, statistics, etc., for quantum particles with integer spin values, such as

0

,

1

,

2

,

…

,

n

, where

n

is usually thought to be less than

3

, for field quanta such as photons, gravitons, gluons, hypothetical Higgs bosons, etc).

For massive quantum particles such as electrons, protons, neutrons, atoms, and so on, the spin

property

has been initially observed for atoms by applying a magnetic field as in the famous Stern-Gerlach experiment, (although the applied field may also be electric or gravitational, (see for example

[

1

]

)). All such spins interact with each other if the spin value is non-zero (i.e., generally, an integer, or half-integer) thus giving rise to “spin networks”, which can be mathematically represented as in

Defintion 0.1

above; in the case of electrons, protons and neutrons such interactions are magnetic dipolar ones, and in an over-simplified, but not a physically accurate ‘picture’, these are often thought of as ‘very tiny magnets–or magnetic dipoles–that line up, or flip up and down together, etc’.

As a practical (and thus ‘intuitive’, pictorial) example, the detection of all
MRI (2D-FT) images (

http://planetmath.org/TwoDimensionalFourierTransforms

) employed in

http://www.wpi.edu/Pubs/ETD/Available/etd-081707-080430/unrestricted/dbennett.pdf

clinical medicine and biomedical research, as well as all (multi-) Nuclear Magnetic Resonance (NMR) spectra employed in physical, chemical, biophyisical/biochemical/biomedical, polymer and agricultural research involves quantum transitions between spin networks or spin foams.

References

1

Werner Heisenberg.

The Physical Principles of Quantum Theory

. New York: Dover Publications, Inc.(1952), pp.39-47.

2

F. W. Byron, Jr. and R. W. Fuller.

Mathematical Principles of Classical and Quantum Physics.

, New York: Dover Publications, Inc. (1992).

3

Baez, J. & Dolan, J., 1998a, Higher-Dimensional Algebra III.

n-Categories

and the

Algebra

of Opetopes, in

Advances in Mathematics

,

135

: 145-206.

4

Baez, J. & Dolan, J., 1998b,

“Categorification”, Higher Category Theory, Contemporary Mathematics

,

230

, Providence:

AMS

, 1-36.

5

Baez, J. & Dolan, J., 2001, From

Finite Sets

to Feynman Diagrams, in

Mathematics Unlimited – 2001 and Beyond

, Berlin: Springer, pp. 29–50.

Title

spin networks and spin foams

Canonical

name

SpinNetworksAndSpinFoams

Date of creation

2013-03-22 18:14:34

Last modified on

2013-03-22 18:14:34

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

72

Author

bci1 (20947)

Entry type

Topic

Classification

msc 81Q05

Classification

msc 81P05

Classification

msc 81R15

Classification

msc 81-00

Synonym

one-dimensional

C

⁢

W

-complexes and category of

C

⁢

W

complexes

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TwoDimensionalFourierTransforms

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IrreducibleRepresentationsOfS_n

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IrreducibleRepresentationsOfTheSpecialLinearGroupOverMathbbF_p

Related topic

TheoremOnCWComplexApproximationOfQuantumStateSpacesInQAT

Related topic

Spinor

Related topic

Qua

Defines

formal spin network

Defines

formal spin foams

Defines

Lie group representation

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