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Quantum Version of Gauge Invariance and Nucleon Internal Structure (dedicate to my late wife) Fan Wang 1 Xiao-Fu L ü 2 , Wei-Min Sun 1 , Xiang-Song Chen 2

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Quantum Version of Gauge Invariance

and

Nucleon Internal Structure

(dedicate to my late wife)

Fan Wang1

Xiao-Fu Lü2, Wei-Min Sun1, Xiang-Song Chen2

1 Dept. of Phys., Nanjing Univ., Nanjing, 210093

[email protected]

2 Dept. of Phys., Sichuan Univ., Chengdu, 610064

[email protected]


Outline
Outline

I. Introduction

II. Quantum version of gauge, gauge transformation and gauge invariance

III.Gauge independent of canonical quark momentum and angular momentum operators

IV.A warning on quantum gauge transformation realized by path integral approach

V. Conclusion


I introduction
I. Introduction

  • In the Faddeev-Popov path integral approach of the quantization of gauge field, the gauge invariance directly follows the classical version.

  • In the canonical quantization the gauge invariance is distinguished to be strictly gauge invariance, which corresponds to the classical version, gauge invariance, weak gauge invariance, and gauge independent. (F.Strocchi and A.S.Wightman, J. Math. Phys., 15, 2198 (1974))

  • Physically the gauge independent local Hermitian operator corresponds to observable. Therefore the quantum version of gauge field includes observables other than the classically gauge invariant ones.

  • Path integral calculation should be combined with canonical quantization otherwise will lead to error. (Hung Cheng and Er-Cheng Tsai, Phys. Rev. Lett. 57, 511(1987); W.M.Sun, X.S.Chen and F. Wang, Phys. Lett. B483, 299(2000); B503, 430(2001), B569,211(2003),

    J. Phys. G25, 2021(1999).


Classical gauge invariance has been emphasized in nucleon structure study
Classical gauge invariance has been emphasized in nucleon structure study

In the nucleon internal structure studies, there seems to be

an attitude that only classically gauge invariant operators are

used to describe the nucleon internal structure.

For example one uses the gauge invariant covariant

derivative operator, the mechanical momentum operator,

instead of the gauge non-invariant canonical momentum

operator to describe the quark momentum distribution;

In the nucleon spin structure studies, one searches the

gauge invariant quark angular momentum operator rather

than the canonical angular momentum operator, which is not

gauge invariant.

A gauge invariant gluon spin operator has been searched

for a long time.


Gauge invariant ones do not satisfy the canonical commutation relations
Gauge invariant ones do not satisfy the canonical commutation relations

  • In fact the gauge invariant quark momentum and angular momentum operators do not satisfy the canonical commutation relation, they are not the momentum and angular momentum operators which one used in quantum physics.

  • This misuse will ruin the partial wave analysis and the multi-pole radiation analysis which is so widely used in atomic, molecular, nuclear and hadron spectroscopy because the canonical momentum and angular momentum are used in such a decomposition.

  • If possible one should come back to canonical quark momentum and angular momentum and we will show it is.


II.Quantum commutation relations version of Gauge, Gauge transformation, Gauge Invariance

A gauge of Interacting gauge field is defined as

(a)

Operator fields are defined in Hilbert space H

and

in H there is a representation U of the Poincare group.

(b)

An indefinite metric form < · , · >, first introduced by Gupta-Bleuler, is defined in H with respect to it the representation U is unitary.

(c)

There is a subspace H’ of H which has the following properties,

(1)

The restriction of the indefinite metric form <·,·> to H’ is bounded and nonnegative

forΨ∈ H’


(2) commutation relations

There is a common dense domain D H’ for all local observables O such that OD D. Here O is an operator of observables and in this subspace

where

(3)

The subset of H’’ consist of all vectors of H’ with zero length and physical space is defined by the quotient space .

There exists an unique vector , called vacuum which is invariant under the Poincare group.

……


A quantum gauge transformation commutation relations

is an ordered pair consisting of two gauges

together with a bijection g of

such that

for all ,

with ,

and .


  • The classical version commutation relations

    and

    consist of the special gauge transformation. It is far

    from true that all quantum gauge transformations

    can be expressed in this restricted version.

    Even the usual Coulomb gauge can not be obtained

    from the covariant gauge through such a special

    gauge transformation.

    (O.Steinmann, hep-th/0411095)


Gauge invariance is distiguished to be
Gauge invariance is distiguished to be commutation relations

  • An operator O, mapping H into H, is gauge independent if

    for all and any

  • An operator O, mapping H into H, is weakly gauge invariant if it and its adjoint leave H’’ invariant:

  • An operator O, mapping H into H, is gauge invariant if it and its adjoint leave H’ invariant:

  • An operator O, mapping H into H, is strictly gauge invariant if it is gauge invariant and commutes with


III Gauge Independent of canonical quark momentum and angular momentum operators

Let’s repeat that In the quantum gauge field theory not only the strict gauge invariant, i.e., classically gauge invariant, but also the gauge invariant, weak gauge invariant even gauge independent operators are observables, their matrix elements between physical states are gauge independent.

F.Strocchi and A.S. Wightman, J. Math. Phys., 15, 2218(1974)


A gauge independent operator is defined by angular momentum operators

Where

and

This means that the matrix elements depend on the equivalent classes

All equivalent classes constitute the quotient space


Associated with a local self adjoint operator O defined on H there is a uniquely determined operator in physical space H’/H’’, whose matrix elements only depends on the equivalent class and

This fact means that one has defined an operator

in the Hilbert space of physical states

It has to be emphasized that there are two restrictions for an observable, one is that the states must be the physical states, the other is that the operator must be gauge independent


  • We show that they are gauge independent operators perturbatively

  • All of the zero mode states of H’’ can be represented by the polynomialsacting on the physical states i.e.

where are the arbitrary constants.


In order to prove an operator to be gauge independent it is enough to show that

And

where

In the case of the quark momentum operator we have to study the matrix elements

where S is the S matrix.


By using the perturbation expansion it can be rewritten as enough to show that

where

since

the every term in the sum of the above Eq. can be rewritten in the form



Thus the Eq. implies that used.

By the similar method it is easy to prove that

This argument obviously can be directly extend to the angular momentum operator and so it is also gauge independent.

The condition is no explicit gluon component in the physical states.


Iv a warning on the quantum gauge transformation realized by path integral approach
IV.A warning on the quantum gauge transformation realized by path integral approach

There are discussions of quantum gauge transformation realized by path integral approach taking the advantage of path integral formalism where all fields are treated as classical c numbers. One can use the classical gauge transformation to realize the quantum gauge transformation.


  • Suppose the F, gauge correspond to covariant gauge and Coulomb gauge, the above transformation seems to realize a quantum gauge transformation from covariant gauge to Coulomb gauge.

    However we already know that the special gauge transformation can transform a covariant gauge to a “Coulomb type gauge” but never to a real Coulomb gauge because the Maxwell Eq. is not satisfied in covariant gauge but do satisfied in Coulomb gauge as an operator Eq.


  • Warning Coulomb gauge, the above transformation seems to realize a quantum gauge transformation from covariant gauge to Coulomb gauge.

    This is a new example that when one use the path integral results the canonical quantization results should be consulted. A naive use might lead to error.


V conclusion
V.Conclusion Coulomb gauge, the above transformation seems to realize a quantum gauge transformation from covariant gauge to Coulomb gauge.

  • Canonical quark momentum and angular momentum are gauge independent operators and so are observables.

  • One’d better use the canonical momentum and angular momentum to study the nucleon internal structure.

  • One has already got information of the canonical quark orbital angular from hadron spectroscopy. The direct measurement of quark orbital angular momentum should be studied.

  • Canonical quantization results should be consulted in the path integral calculation.


Thanks Coulomb gauge, the above transformation seems to realize a quantum gauge transformation from covariant gauge to Coulomb gauge.

for your patient


IV A warning of quantum gauge transformation by Coulomb gauge, the above transformation seems to realize a quantum gauge transformation from covariant gauge to Coulomb gauge.

path integral approach

Let’s discuss a calculation of the matrices between the physical states, which are created by gauge invariant operators O acting on the vacuum state, in two different gauge by path integral approach. Let’s start from the covariant gauge, for the momentum and angular momentum operators we shall calculate the generalized Green’s functions

(a)

and


(b) Coulomb gauge, the above transformation seems to realize a quantum gauge transformation from covariant gauge to Coulomb gauge.

From Eqs. (a),(b) one can obtain the corresponding matrix elements as

and



Let’s make a special gauge transformation from covariant gauge to Coulomb type gauge to obtain the momentum and angular momentum operators,

and

Where ω is a gauge transformation from the covariant gauge to the Coulomb gauge i.e. Is the Coulomb gauge quark field operator.


The matrix elements can also be calculated in the gauge to Coulomb type gauge to obtain the momentum and angular momentum operators,Coulomb gauge

where a Faddeev identity has been inserted.

Since there are two δfunction, and

ω is determined by the condition that it transforms

to

This implies that ω is a gauge transformation from the Coulomb gauge to covariant gauge. Then Generalized Green function can be transformed from Coulomb gauge to covariant gauge,


Few comments about this gauge transformation: gauge to Coulomb type gauge to obtain the momentum and angular momentum operators,

1.Canonical quantization shows that Coulomb gauge can not obtained from covariant gauge through a special gauge transformation, so this path integral gauge transformation is spurious.

2.For gauge non-invariant operators, the change of integral order of [dω] and other part is illegal.

Such kind path integral formula of gauge transformation for gauge non-invariant operators are not reliable.


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