aarjaneiro/planetmath huggingface dataset:
rigged Hilbert space

rigged Hilbert space

In

extensions

of Quantum Mechanics

[

1

,

2

]

, the

concept

of

rigged Hilbert spaces

allows one “to put together” the discrete spectrum of

eigenvalues

corresponding to the bound states (

eigenvectors

) with the

continuous spectrum

(as , for example, in the case of the ionization of an atom or the photoelectric effect).

Definition 0.1

.

A

rigged Hilbert space

is a pair

(

ℋ

,

ϕ

)

with

ℋ

a

Hilbert space

and

ϕ

is a dense subspace with a

topological vector space

structure

for which the

inclusion map

i

is

continuous

. Between

ℋ

and its

dual space

ℋ

*

there is defined the

adjoint map

i

*

:

ℋ

*

→

ϕ

*

of the continuous inclusion map

i

. The duality

pairing

between

ϕ

and

ϕ

*

also needs to be

compatible

with the

inner product

on

ℋ

:

⟨

u

,

v

⟩

ϕ

×

ϕ

*

=

(

u

,

v

)

ℋ

whenever

u

∈

ϕ

⊂

ℋ

and

v

∈

ℋ

=

ℋ

*

⊂

ϕ

*

.

References

1

R. de la Madrid, “The role of the rigged Hilbert space in Quantum Mechanics.”, Eur. J. Phys. 26, 287 (2005);

q

⁢

u

⁢

a

⁢

n

⁢

t

-

p

⁢

h

/

0502053

.

2

J-P. Antoine, “Quantum Mechanics Beyond Hilbert Space” (1996), appearing in

Irreversibility and Causality, Semigroups and Rigged Hilbert Spaces

, Arno Bohm, Heinz-Dietrich Doebner, Piotr Kielanowski, eds., Springer-Verlag,

I

⁢

S

⁢

B

⁢

N

⁢

3

-

540

-

64305

-

2

.

Title

rigged Hilbert space

Canonical name

RiggedHilbertSpace

Date of creation

2013-03-22 19:22:48

Last modified on

2013-03-22 19:22:48

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

6

Author

bci1 (20947)

Entry type

Definition

Classification

msc 81Q20

Synonym

Gelfand triple

Defines

dual Hilbert space

Defines

adjoint map

Defines

eigen spectrum

Generated on Sat Feb 10 13:05:31 2018 by

LaTeXML
Jensen’s inequality

Jensen’s inequality

If

f

is a

convex function

on the

interval

[

a

,

b

]

, for each

{

x

k

}

k

=

1

n

∈

[

a

,

b

]

and each

{

μ

k

}

k

=

1

n

with

μ

k

≥

0

one has:

f

⁢

(

∑

k

=

1

n

μ

k

⁢

x

k

∑

k

n

μ

k

)

≤

∑

k

=

1

n

μ

k

⁢

f

⁢

(

x

k

)

∑

k

n

μ

k

.

A common situation occurs when

μ

1

+

μ

2

+

⋯

+

μ

n

=

1

; in this case, the

inequality

simplifies to:

f

⁢

(

∑

k

=

1

n

μ

k

⁢

x

k

)

≤

∑

k

=

1

n

μ

k

⁢

f

⁢

(

x

k

)

where

0

≤

μ

k

≤

1

.

If

f

is a

concave function

, the inequality is reversed.

Example:

f

⁢

(

x

)

=

x

2

is a convex function on

[

0

,

10

]

.
Then

(

0.2

⋅

4

+

0.5

⋅

3

+

0.3

⋅

7

)

2

≤

0.2

⁢

(

4

2

)

+

0.5

⁢

(

3

2

)

+

0.3

⁢

(

7

2

)

.

A very special case of this inequality is when

μ

k

=

1

n

because then

f

⁢

(

1

n

⁢

∑

k

=

1

n

x

k

)

≤

1

n

⁢

∑

k

=

1

n

f

⁢

(

x

k

)

that is, the value of the

function

at the mean of the

x

k

is less or equal than the mean of the values of the function at each

x

k

.

There is another formulation of Jensen’s inequality used in

probability

:

Let

X

be some

random variable

, and let

f

⁢

(

x

)

be a convex function (defined at least on a segment containing the range of

X

). Then the

expected value

of

f

⁢

(

X

)

is at least the value of

f

at the mean of

X

:

E

⁢

[

f

⁢

(

X

)

]

≥

f

⁢

(

E

⁢

[

X

]

)

.

With this approach, the weights of the first form can be seen as probabilities.

Title

Jensen’s inequality

Canonical name

JensensInequality

Date of creation

2013-03-22 11:46:30

Last modified on

2013-03-22 11:46:30

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

13

Author

Andrea Ambrosio (7332)

Entry type

Theorem

Classification

msc 81Q30

Classification

msc 26D15

Classification

msc 39B62

Classification

msc 18-00

Related topic

ConvexFunction

Related topic

ConcaveFunction

Related topic

ArithmeticGeometricMeansInequality

Related topic

ProofOfGeneralMeansInequality

Generated on Sat Feb 10 13:05:35 2018 by

LaTeXML
quantum nano-automata

quantum nano-automata

\xyoption

curve

Definition 0.1

.

A

quantum nano-automaton

or

(quantum nano-computer)

is defined as a microphysical nano-device represented by a

quantum automaton

that is also capable of quantum computation involving, for example, a field of computations specified over a

quantum groupoid

.
The latter can be realized as a

locally compact

(topological)

groupoid

, topological semi-group;in the more general
case, it can be defined over a

quantum

algebraic category

,

quantum algebroid

or a higher dimensional

algebraic structure

(as previously defined in

higher dimensional algebra

).

Note:

Quantum nano-automata possess intrinsic, extended quantum symmetries defined by quantum groupoid or quantum algebroid

representations

Title

quantum nano-automata

Canonical name

QuantumNanoautomata

Date of creation

2013-03-22 18:11:54

Last modified on

2013-03-22 18:11:54

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

14

Author

bci1 (20947)

Entry type

Topic

Classification

msc 81Q60

Classification

msc 81Q50

Classification

msc 68Q85

Classification

msc 68Q05

Classification

msc 81Q30

Synonym

nanoelectronics

Synonym

nanotechnology computing devices

Related topic

QuantumAutomataAndQuantumComputation2

Defines

quantum nano-automaton

Defines

quantum nano-computer

Generated on Sat Feb 10 13:05:56 2018 by

LaTeXML
bibliography for group theory

bibliography for group theory

References for

Group Theory

The following are excellent sources for the indicated areas in Group Theory.

Foundations, MSC 20A

1.

David S. Dummit, Richard M. Foote,

Abstract Algebra

, John Wiley and Sons, Inc. 1999

2.

I. N. Herstein,

Topics in

Algebra

, Xerox College Publishing, 1975.

A classic undergraduate textbook introducing groups, rings, fields, and

linear algebra

.

3.

Serge Lang,

Algebra

, Addison-Wesley Publishing Company, 1997.

A textbook

covering

a broad range of topics in algebra; some emphasis is placed on the parts of algebra most useful in

algebraic geometry

. In particular, basic

homological algebra

and basic

category theory

are well covered.

4.

John S. Rose,

A Course on Group Theory

, Dover Publications, New York, 1994.

5.

Thomas W. Hungerford,

Algebra

, Springer-Verlag Graduate Texts in Mathematics 73, 1974.

Covers all of algebra from a more classical viewpoint; when dealing with rings, the focus is on noncommutative rings. Only two chapters on group theory are present. A brief discussion of

categories

is included.

6.

Joseph J. Rotman,

An Introduction to the Theory of Groups

, fourth edition, Springer-Verlag Graduate Texts in Mathematics 148, 1994.

A thorough introduction to the theory of groups. Emphasis is on

groups with

no further structure; includes a chapter on computability issues and the

word problem

.

Applications and books on line, MSC: 81R05,81R10

1.

Predrag Cvitanovic,

Group Theory

,

http://planetmath.org/?op=getobj&from=books&id=60

on line

2.

Michael Tinkham,

Group Theory and Quantum Mechanics

,

http://planetmath.org/?op=getobj&from=books&id=183

on line

3.

Morton Hamermesh,

Group Theory and Physical Applications

,

http://planetmath.org/?op=getobj&from=books&id=182

on line

4.

Emil Artin

Galois Theory

http://planetmath.org/?op=getobj&from=books&id=25

on line

Title

bibliography for group theory

Canonical name

BibliographyForGroupTheory

Date of creation

2013-03-22 14:14:24

Last modified on

2013-03-22 14:14:24

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

15

Author

alozano (2414)

Entry type

Bibliography

Classification

msc 81R05

Classification

msc 81R10

Classification

msc 20-00

Generated on Sat Feb 10 13:05:57 2018 by

LaTeXML
C^*-algebra homomorphisms are continuous

C

*

-algebra homomorphisms are continuous

Theorem -

Let

𝒜

,

ℬ

be

C

*

-

algebras

(

http://planetmath.org/CAlgebra

) and

f

:

𝒜

⟶

ℬ

a *-

homomorphism

. Then

f

is bounded (

http://planetmath.org/ContinuousLinearMapping

) and

∥

f

∥

≤

1

(where

∥

f

∥

is the norm (

http://planetmath.org/OperatorNorm

) of

f

seen as a

linear operator

between the spaces

𝒜

and

ℬ

).

For this reason it is often said that homomorphisms between

C

*

-algebras are automatically

continuous

(

http://planetmath.org/ContinuousLinearMapping

).

Corollary -

A *-

isomorphism

between

C

*

-algebras is an isometric isomorphism (

http://planetmath.org/IsometricIsomorphism

).

Proof of Theorem :

Let us first suppose that

𝒜

and

ℬ

have

identity elements

, both denoted by

e

.

We denote by

σ

⁢

(

x

)

and

R

σ

⁢

(

x

)

the

spectrum

and the

spectral radius

of an element

x

∈

𝒜

or

ℬ

.

Let

a

∈

𝒜

and

λ

∈

ℂ

. If

a

-

λ

⁢

e

is

invertible

in

𝒜

, then

f

⁢

(

a

-

λ

⁢

e

)

is invertible in

ℬ

. Thus,

σ

⁢

(

f

⁢

(

a

)

)

⊆

σ

⁢

(

a

)

.

Hence

R

σ

⁢

(

f

⁢

(

a

)

)

≤

R

σ

⁢

(

a

)

for every

a

∈

𝒜

. Therefore, by the result from this entry (

http://planetmath.org/NormAndSpectralRadiusInCAlgebras

),

∥

f

⁢

(

a

)

∥

=

R

σ

⁢

(

f

⁢

(

a

)

*

⁢

f

⁢

(

a

)

)

=

R

σ

⁢

(

f

⁢

(

a

*

⁢

a

)

)

≤

R

σ

⁢

(

a

*

⁢

a

)

=

∥

a

∥

.

We conclude that

f

is and

∥

f

∥

≤

1

.

If

𝒜

or

ℬ

do not have identity elements, we can consider their

minimal unitizations

, and the result follows from the above .

□

Proof of Corollary :

This follows from the fact that

f

-

1

is also a *-homomorphism and therefore

∥

f

-

1

⁢

(

b

)

∥

≤

∥

b

∥

for every

b

∈

ℬ

.

□

Title

C

*

-algebra homomorphisms are continuous

Canonical name

CalgebraHomomorphismsAreContinuous

Date of creation

2013-03-22 17:40:06

Last modified on

2013-03-22 17:40:06

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

14

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 81R15

Classification

msc 46L05

Synonym

automatic

continuity

of

C

*

-homomorphisms

Synonym

homomorphisms of

C

*

-algebras are continuous

Related topic

ContinuousLinearMapping

Related topic

OperatorNorm

Related topic

C_cG

Related topic

UniformContinuityOverLocallyCompactQuantumGroupoids

Related topic

CAlgebra

Related topic

CAlgebra3

Related topic

NormAndSpectralRadiusInCAlgebras

Related topic

EquivalenceOfDefinitionsOfCAlgebra

Related topic

GroupoidCConvolutionAlgebra

Defines

automatically continuous homomorphism of

C

*

–algebras

Generated on Sat Feb 10 13:06:00 2018 by

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Cartan calculus

Cartan calculus

Suppose

M

is a

smooth manifold

, and denote by

Ω

⁢

(

M

)

the

algebra

of

differential forms

on

M

. Then, the

Cartan calculus

consists of the following three types of

linear operators

on

Ω

⁢

(

M

)

:

1.

the

exterior derivative

d

,

2.

the space of

Lie derivative

operators

ℒ

X

, where

X

is a

vector field

on

M

, and

3.

the space of contraction operators

ι

X

, where

X

is a vector field on

M

.

The above operators

satisfy

the following

identities

for any vector fields

X

and

Y

on

M

:

d

2

=

0

,

(1)

d

⁢

ℒ

X

-

ℒ

X

⁢

d

=

0

,

(2)

d

⁢

ι

X

+

ι

X

⁢

d

=

ℒ

X

,

(3)

ℒ

X

⁢

ℒ

Y

-

ℒ

Y

⁢

ℒ

X

=

ℒ

[

X

,

Y

]

,

(4)

ℒ

X

⁢

ι

Y

-

ι

Y

⁢

ℒ

X

=

ι

[

X

,

Y

]

,

(5)

ι

X

⁢

ι

Y

+

ι

Y

⁢

ι

X

=

0

,

(6)

where the brackets on the right hand side denote the Lie bracket of vector fields.

The identity (

3

) is known as

Cartan’s magic

formula

or

Cartan’s identity

Interpretation as a Lie Superalgebra

Since

Ω

⁢

(

M

)

is a graded algebra, there is a natural grading on the space of linear operators on

Ω

⁢

(

M

)

. Under this grading, the exterior derivative

d

is degree

1

, the Lie derivative operators

ℒ

X

are degree

0

, and the contraction operators

ι

X

are degree

-

1

.

The identities (

1

)-(

6

) may each be written in the form

A

⁢

B

±

B

⁢

A

=

C

,

(7)

where a

plus sign

is used if

A

and

B

are both of odd degree, and a minus sign is used otherwise. Equations of this form are called

supercommutation relations

and are usually written in the form

[

A

,

B

]

=

C

,

(8)

where the bracket in (

8

) is a

Lie superbracket

. A Lie superbracket is a

generalization

of a

Lie bracket

.

Since the Cartan Calculus operators are

closed under

the Lie superbracket, the

vector space

spanned by the Cartan Calculus operators has the

structure

of a

Lie superalgebra

.

Graded derivations of

Ω

⁢

(

M

)

Definition 1

.

A degree

k

linear operator

A

on

Ω

⁢

(

M

)

is a

graded derivation

if it satisfies the following

property

for any

p

-form

ω

and any differential form

η

:

A

⁢

(

ω

∧

η

)

=

A

⁢

(

ω

)

∧

η

+

(

-

1

)

k

⁢

p

⁢

ω

∧

A

⁢

(

η

)

.

(9)

All of the Calculus operators are graded derivations of

Ω

⁢

(

M

)

.

Title

Cartan calculus

Canonical

name

CartanCalculus

Date of creation

2013-03-22 15:35:39

Last modified on

2013-03-22 15:35:39

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

14

Author

bci1 (20947)

Entry type

Definition

Classification

msc 81R15

Classification

msc 17B70

Classification

msc 81R50

Classification

msc 53A45

Classification

msc 81Q60

Classification

msc 58A15

Classification

msc 14F40

Classification

msc 13N15

Synonym

Lie superalgebra

Related topic

LieSuperalgebra3

Related topic

LieDerivative

Related topic

DifferentialForms

Defines

anticommutator bracket

Defines

Cartan’s magic formula

Defines

supercommutation relation

Defines

graded derivation

Generated on Sat Feb 10 13:06:02 2018 by

LaTeXML
compact quantum group

compact quantum group

Definition 0.1

.

A

compact quantum group

,

Q

C

⁢

G

is defined as a particular case of a

locally compact quantum group

Q

L

⁢

C

⁢

G

, that is, a quadruple

(

A

,

Δ

,

μ

,

ν

)

, where

A

is either a

C

*

– or a

W

*

–

algebra

equipped with a co-associative comultiplication (

http://planetmath.org/WeakHopfCAlgebra2

)

Δ

:

A

→

A

⊗

A

, and two

faithful

semi-finite normal weights,

μ

and

ν

–right and -left Haar measures, and also when the object space

𝐎

of the latter

Q

L

⁢

C

⁢

G

is replaced by a

compact

topological space

Q

C

⁢

G

T

, instead of being a

locally compact topological space

like

Q

L

⁢

C

⁢

G

.

References

1

A. Maes, and A. VanDaele. 1998.

http://arxiv.org/PS_cache/math/pdf/9803/9803122v1.pdf

Notes on Compact Quantum Groups.,

a

⁢

r

⁢

x

⁢

i

⁢

v

.

o

⁢

r

⁢

g

.

m

⁢

a

⁢

t

⁢

h

-

F

⁢

A

-

9803122

⁢

v

⁢

1

, 43 pp.

Title

compact quantum group

Canonical name

CompactQuantumGroup

Date of creation

2013-03-22 18:24:07

Last modified on

2013-03-22 18:24:07

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

25

Author

bci1 (20947)

Entry type

Definition

Classification

msc 81R15

Classification

msc 81R50

Classification

msc 46L05

Synonym

quantum group

Synonym

compact matrix quantum group

Related topic

CAlgebra3

Related topic

QuantumOperatorAlgebrasInQuantumFieldTheories

Related topic

FiniteQuantumGroup

Related topic

DualityInMathematics

Related topic

LocallyCompactQuantumGroup

Related topic

QuantumGroups

Related topic

GelfandTransform

Generated on Sat Feb 10 13:06:04 2018 by

LaTeXML
locally compact quantum group

locally compact quantum group

Definition 0.1

.

A

locally compact quantum group

defined as in ref.

[

1

]

is a

quadruple

G

=

(

A

,

Δ

,

μ

,

ν

)

, where

A

is either a

C

*

– or a

W

*

–

algebra

equipped with a co-associative comultiplication (

http://planetmath.org/WeakHopfCAlgebra2

)

Δ

:

A

→

A

⊗

A

and two

faithful

semi-finite normal weights,

μ

and

ν

–

right and -left Haar measures

.

0.0.1

Examples

1.

An ordinary

unimodular group

G

with Haar measure

μ

.

A

:=

L

∞

⁢

(

G

,

μ

)

,

Δ

:

f

⁢

(

g

)

↦

f

⁢

(

g

⁢

h

)

,

S

:

f

⁢

(

g

)

↦

f

⁢

(

g

-

1

)

,

ϕ

⁢

(

f

)

=

∫

G

f

⁢

(

g

)

⁢

𝑑

μ

⁢

(

g

)

, where

g

,

h

∈

G

,

f

∈

L

∞

⁢

(

G

,

μ

)

.

2.

A:= Ł(G) is the

von Neumann algebra

generated by left-translations

L

g

or by left convolutions

L

f

:=

∫

G

f

⁢

(

g

)

⁢

L

g

⁢

𝑑

μ

⁢

(

g

)

with

continuous functions

f

(

.

)

∈

L

1

(

G

,

μ

)

Δ

:

↦

L

g

⊗

L

g

↦

L

g

-

1

,

ϕ

(

f

)

=

f

(

e

)

, where

g

∈

G

, and e is the unit of G.

References

1

Leonid Vainerman. 2003.

http://planetmath.org/?op=getobj&from=papers&id=471

Locally Compact Quantum Groups and

Groupoids

: Proceedings of the Meeting of Theoretical Physicists and Mathematicians, Strasbourg, February 21-23, 2002.,

Series in Mathematics and Theoretical Physics

,

2

, Series ed. V. Turaev., Walter de Gruyter Gmbh & Co: Berlin.

Title

locally compact quantum group

Canonical name

LocallyCompactQuantumGroup

Date of creation

2013-03-22 18:21:24

Last modified on

2013-03-22 18:21:24

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

18

Author

bci1 (20947)

Entry type

Definition

Classification

msc 81R50

Classification

msc 46M20

Classification

msc 18B40

Classification

msc 22A22

Classification

msc 17B37

Classification

msc 46L05

Classification

msc 22D25

Synonym

Hopf algebras

Synonym

ring groups

Related topic

CompactQuantumGroup

Related topic

LocallyCompactQuantumGroupsUniformContinuity2

Related topic

RepresentationsOfLocallyCompactGroupoids

Related topic

VonNeumannAlgebra

Related topic

WeakHopfCAlgebra2

Related topic

LocallyCompactHausdorffSpace

Related topic

QuantumGroups

Defines

quantum group

Defines

local quantum symmetry

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quantum space-times

quantum space-times

This is a contributed topic on

quantum space-times

viewed from general relativistic and quantum gravity standpoints,
and includes, for example,

quantum geometry

fundamental notions.

1

Quantum Space-times (QST)

The concept of

quantum space-times (QST)

is fundamental to the

development

of relativistic

quantum theories

and at this point it can only be broadly defined as a

class of mathematical spaces that allow the construction of quantum physical theories in a manner consistent with both relativistic principles and quantum gravity

.
There is no

universal

agreement amongst either theoretical physicists or mathematicians who
work on Physical Mathematics about either a specific definition of such quantum space-times or
how to develop a valid classification theory of quantum space-times. However, several specific
definitions or models were proposed and a list of such examples is presented next.

1.1

Specific Definitions for Models of Quantum Space-Times (QSTs) and Quantum Geometry

•

QSTs represented by posets or causal sets

•

QSTs represented by so-called quantum topoi (QTs) with Heyting

logic algebra

as a

subobject classifier

•

QSTs represented by Topological Quantum Field Theories (TQFTs) or

Homotopy

QFTs

•

QSTs represented as
spin foams of spin networks (

http://planetmath.org/SpinNetworksAndSpinFoams

)

•

QSTs represented as a

noncommutative

, algebraic– and/or “geometrical”–quantized space as in

noncommutative geometry

models for SUSY

•

QSTs represented as generalized

Riemannian manifolds

with quantum

tangent spaces

•

QSTs represented as

presheaves

of local nets of quantum operators in Algebraic QFT

•

QSTs represented as quantum fields (QF) on a (physical) lattice of geometric points

•

QSTs represented as consisting of quantum loops

•

QSTs represented as fractal dimension spaces

•

QST represented as a

quantum category

of

locally compact

grpoupoids

•

QSTs represented as

categories

or spaces of quantized strings as in string theories

•

Twistor

representations

in quantum gravity (QG) (introduced by Sir Roger Penrose).

Title

quantum space-times

Canonical name

QuantumSpacetimes

Date of creation

2013-03-22 18:15:56

Last modified on

2013-03-22 18:15:56

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

28

Author

bci1 (20947)

Entry type

Topic

Classification

msc 81R60

Classification

msc 81R50

Classification

msc 81P10

Classification

msc 83C47

Classification

msc 83C75

Classification

msc 83C45

Classification

msc 81P05

Synonym

QST

Synonym

quantum state spaces

Synonym

quantum spaces

Related topic

QuantumFundamentalGroupoids

Related topic

QuantumGroupoids2

Related topic

QuantumGroups

Related topic

QuantumSystem

Related topic

QEDInTheoreticalAndMathematicalPhysics

Related topic

HamiltonianOperatorOfAQuantumSystem

Related topic

QuantumChromodynamicsQCD

Related topic

QuantumAutomataAndQuantumComputation2

Related topic

Quantization

Related topic

QEDInTheoreticalAndMathema

Defines

quantum space-time structure

Defines

QST

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Feynman path integral

Feynman path integral

A generalisation of multi-dimensional

integral

, written

∫

𝒟

⁢

ϕ

⁢

exp

⁢

(

ℱ

⁢

[

ϕ

]

)

where

ϕ

ranges over some restricted set of

functions

from a

measure space

X

to some space with reasonably nice

algebraic structure

. The simplest example is the case where

ϕ

∈

L

2

⁢

[

X

,

ℝ

]

and

F

⁢

[

ϕ

]

=

-

π

⁢

∫

X

ϕ

2

⁢

(

x

)

⁢

𝑑

μ

⁢

(

x

)

in which case it can be argued that the result is

1

. The

argument

is by

analogy

to the

Gaussian integral

∫

ℝ

n

𝑑

x

1

⁢

⋯

⁢

𝑑

x

n

⁢

e

-

π

⁢

∑

x

j

2

≡

1

. Alas, one can absorb the

π

into the

measure

on

X

.
Alternatively, following Pierre Cartier and others, one can use this analogy to

define

a measure on

L

2

and proceed axiomatically.

One can bravely trudge onward and hope to come up with something, say à la

Riemann integral

, by partitioning

X

,
picking some representative of each

partition

, approximating the

functional

F

based on these
and calculating a multi-dimensional integral as usual over the sample values of

ϕ

. This leads to some integral

∫

⋯

⁢

𝑑

ϕ

⁢

(

x

1

)

⁢

⋯

⁢

𝑑

ϕ

⁢

(

x

n

)

⁢

e

f

⁢

(

ϕ

⁢

(

x

1

)

,

…

,

ϕ

⁢

(

x

n

)

)

.

One hopes that taking successively finer partitions of

X

will give a

sequence

of integrals which

converge

on some nice limit. I believe Pierre Cartier has shown that this doesn’t usually happen, except for the

trivial

kind of example given above.

The Feynman path integral was constructed as part of a re-formulation of by Richard Feynman, based on the sum-over-histories

postulate

of quantum mechanics, and can be thought of as an adaptation of Green’s function methods for solving initial/boundary value problems. No appropriate measure has been found for this integral and attempts at pseudomeasures have given mixed results.

Remark:

Note however that in solving

quantum field theory

problems one attacks the problem in the Feynman approach by
‘dividing’ it

via

Feynman diagrams that are directly related to specific quantum interactions;
adding the contributions from such Feynman diagrams leads to high precision approximations to the final physical
solution which is finite and physically meaningful, or observable.

References

1

Hui-Hsiung Kuo,

Introduction

to Stochastic Integration

. New York: Springer (2006): 250 - 253

2

J. B. Keller & D. W. McLaughlin, “The Feynman Integral”

Amer. Math. Monthly

82

5 (1975): 451 - 465

Title

Feynman path integral

Canonical name

FeynmanPathIntegral

Date of creation

2013-03-22 12:41:45

Last modified on

2013-03-22 12:41:45

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

19

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 81S40

Related topic

LpSpace

Related topic

RichardFeynman

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