quantum field state on the tetrahedron

quantum field state on the tetrahedron

1

Topological Quantum Field (TQFT) State on the Tetrahedron

Let us consider first a

regular tetrahedron

(

http://planetmath.org/

Tetrahedron

) whose corners will
have attached to them the TQFT symbols representing a TQF state in terms of so-called ‘6j-symbols’ as further detailed next. The vertices of the tetrahedron are located at the points

(

a

x

,

a

y

,

a

z

)

,

(

b

x

,

b

y

,

b

z

)

,

(

c

x

,

c

y

,

c

z

)

, and

(

d

x

,

d

y

,

d

z

)

, that will be labeled, respectively, as

1

,

2

,

3

,

4

.

Definition 1.1

.

A

quantum field (QF) state

ϕ

provides a

total order

denoted by

≤

ϕ

on the
vertices of the tetrahedron, and thus also assigns a ‘direction’ to each edge of the tetrahedron–from the
apparently ‘smaller’ to the apparently ‘larger’ vertices; a QF state also labels each edge

e

=

(

i

,

j

)

,
by an element

ϕ

1

⁢

(

e

)

of

B

A

, which is a

distinguished basis of a fusion

algebra

𝒜

, that is, a finite-dimensional, unital, involutive algebra over

ℂ

–the field of

complex numbers

. Moreover, the QF state assigns an element

ϕ

2

⁢

(

f

)

–called an intertwiner– of a

Hilbert space

ℋ

ϕ

⁢

(

f

)

=

ℋ

ϕ

1

⁢

(

i

⁢

k

)

ϕ

1

⁢

(

j

⁢

k

)

,

ϕ

1

⁢

(

i

⁢

j

)

to each face

f

=

(

i

⁢

j

⁢

k

)

of the tetrahedron, such that

i

≺

ϕ

j

≺

ϕ

k

.

Notes:

A

topological quantum field theory

(

TQFT

) is described as a mathematical approach to

quantum field theory

that allows the computation of

topological invariants

of

quantum state spaces

(QSS), usually for cases of lower

dimensions

encountered in certain condensed phases or strongly correlated (quantum) superfluid states. TQFT has some of its origins in theoretical physics as well as Michael Atiyah (

http://planetmath.org/MichaelFrancisAtiyah

)’s research; this was followed by Edward Witten, Maxim Kontsevich, Jones and Donaldson, who all have been awarded Fields Medals for work related to topological quantum field theory; furthermore, Edward Witten and Maxim Kontsevich shared in 2008 the Crafoord prize for TQFT related work. As an example, Maxim Kontsevich introduced the concept of homological mirror (quantum)

symmetry

in

relation

to a mathematical conjecture in superstring theory.

References

1

V. Kodyiyalam and V. S. Sunder. 2001.

http://planetmath.org/?op=getobj&from=books&id=174

Topological Quantum Field Theories From Subfactors

., Chapman and Hall/CRC.

Title

quantum field state on the tetrahedron

Canonical name

QuantumFieldStateOnTheTetrahedron

Date of creation

2013-03-22 18:22:30

Last modified on

2013-03-22 18:22:30

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

37

Author

bci1 (20947)

Entry type

Definition

Classification

msc 46H35

Classification

msc 81T45

Classification

msc 57R56

Synonym

state on the tetrahedron

Related topic

Tetrahedron

Related topic

QFT

Related topic

Grafix

Related topic

FoundationsOfQuantumFieldTheories

Related topic

MonoidalCategory

Related topic

CrafoordPrize

Defines

quantum field state on a tetrahedron

Defines

TQFT

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