Gel’fand triple

Gel’fand triple

1

Gel’fand (or Gelfand) triple

The basic idea is to equip a separable Hilbert space

ℋ

with a dense

Topological Subvector Space

(

T

⁢

V

⁢

S

)

of test functions in such a manner that the dual of the

subspace

of test functions

T

f

enhances the

Hilbert space

ℋ

through embedding into a larger topological subvector space

T

*

; the

elements of

T

*

represent

generalized eigenvectors

for the

continuous spectrum

of normal-and possibly unbounded-linear operators

L

.

One begins the Gelfand construction with a

Banach space

B

, or the more general TVS considered above, and denote the dual TVS of

B

as

B

*

so that

j

:

B

→

ℋ

is an

injective bounded operator

with dense image. One then considers a canonical isomorphism

i

c

:

ℋ

≅

ℋ

*

determined by the

inner product

given by the

Riesz theorem

. The Banach

transpose

, or dual of

j

is the operator

j

*

:

ℋ

*

→

B

*

.

Then, one has a

composition

morphism

k

:

ℋ

→

B

*

as

ℋ

→

ℋ

*

→

B

*

, so that

k

=

j

*

∘

i

c

that can be employed to define the

Gelfand triple

as follows.

Definition 1.1

.

A

Gelfand triple

is defined by the operator sequence:

B

→

ℋ

→

B

*

,

where

k

is the composition morphism

k

:

ℋ

→

B

*

defined by

j

*

∘

i

c

1.1

Example:

An interesting example of a Gel’fand triple which is preserved by the

Fourier transform

(FT) is obtained when

B

=

𝒮

⁢

(

ℛ

n

)

–the

Schwartz space

,

B

*

=

𝒮

t

-the space of tempered Schwartz

distributions

and

ℋ

=

L

2

⁢

(

ℛ

n

)

.

1.2

Acknowledgement

The formalism utilized here followed the

http://ncatlab.org/nlab/show/Gelfand+triple

presentation by Urs Schreiber of the Gel’fand triple at

http://ncatlab.org/nlab/show/HomePage

n

⁢

L

⁢

a

⁢

b

- a ‘steered’ wiki for Mathematics, Physical Mathematics and Philosophy.

1.3

References

J.-P. Antoine. Dirac formalism and

symmetry

problems in quantum mechanics. I. General Dirac formalism. Journal of Mathematical Physics, 10(1):53–69, 1969.

N.Bogoliubov, A.Logunov, and I.Todorov. Introduction to Axiomatic Quantum Field Theory, chapter 1 Some Basic Concepts of

Functional Analysis

4 The Space of States, pages 12–43, 113–128. Benjamin, Reading, Massachusetts, 1975.

R.de la Madrid. Quantum Mechanics in Rigged Hilbert Space Language. PhD thesis, Depertamento de Fisica Teorica Facultad de Ciencias. Universidad de Valladolid, 2001. (available here)

M.Gadella and F.Gomez. A unified mathematical formalism for the dirac formulation of quantum mechanics. Foundations of Physics, 32:815–869, 2002. (available here)

M.Gadella and F.Gomez. On the mathematical basis of the dirac formulation of quantum mechanics. International Journal of Theoretical Physics, 42:2225–2254, 2003.

M.Gadella and F.Gomez. Dirac formulation of quantum mechanics: Recent and new results. Reports on Mathematical Physics, 59:127–143, 2007.

I.M. Gel’fand and N.J. Vilenkin.

Generalized Functions

, vol. 4: Some Applications of

Harmonic Analysis

, volume4, chapter 2-4, pages 26–133. Academic Press, New York, 1964.

I. M. Gel’fand, A. G. Kostyucenko, Expansion in

eigenfunctions

of

differential

and other operators) (Russian), Dokl. Akad. Nauk SSSR (N.S.) 103 (1955), 349–352, MR73136

I. M. Gel’fand, N. Ja. Vilenkin, Generalized functions, vol. 4. Some applications of harmonic analysis. Equipped Hilbert spaces, Fizmatgiz, Moscow, 1961 MR146653, English transl. Acad. Press 1964 MR173945

Ciprian Foiash, Décompositions intégrales des familles spectrales et
semi–spectrales en opérateurs qui sortent de l’espace Hilbertien, Acta Sci. Math. Szeged 20, 1959, 117–155. MR115092

John Roberts, The Dirac bra and ket formalism, J. Mathematical Phys. 7 (1966), 1097–1104, MR216836, Rigged Hilbert spaces in quantum mechanics , Comm. Math. Physics 3, n. 2 (1966), 98–119, MR228243, euclid

J.E. Roberts. The Dirac bra and ket formalism. Journal of Mathematical Physics, 7(6):1097–1104, 1966.

A.R. Marlow. Unified Dirac–Von Neumann formulation of quantum mechanics. I. Mathematical theory. Journal of Mathematical Physics, 6:919–927, 1965.

E.Prugovecki. The bra and ket formalism in extended Hilbert space. J. Math. Phys., 14:1410–1422, 1973.

J.E. Roberts. Rigged Hilbert spaces in quantum mechanics. Commun. math. Phys., 3:98–119, 1966.

Tjostheim. A note on the unified Dirac-Von Neumann formulation of quantum mechanics. Journal of Mathematical Physics, 16(4):766–767, 4 1975

Title

Gel’fand triple

Canonical name

GelfandTriple

Date of creation

2013-03-22 19:22:51

Last modified on

2013-03-22 19:22:51

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

9

Author

bci1 (20947)

Entry type

Topic

Classification

msc 81P10

Synonym

rigged Hilbert space

Defines

𝒮

Defines

𝒮

t

Defines

injective bounded operator

Defines

<

m

x

>

operator

Defines

TVS

Defines

topological subvector space

Defines

canonical isomorphism

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