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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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
J. Phys. G25, 2021(1999).
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
A gauge invariant gluon spin operator has been searched
for a long time.
II.Quantum commutation relations version of Gauge, Gauge transformation, Gauge Invariance
A gauge of Interacting gauge field is defined as
Operator fields are defined in Hilbert space H
in H there is a representation U of the Poincare group.
An indefinite metric form < · , · >, first introduced by Gupta-Bleuler, is defined in H with respect to it the representation U is unitary.
There is a subspace H’ of H which has the following properties,
The restriction of the indefinite metric form <·,·> to H’ is bounded and nonnegative
(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
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
for all ,
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
for all and any
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
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
where are the arbitrary constants.
In order to prove an operator to be gauge independent it is enough to show that
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
the every term in the sum of the above Eq. can be rewritten in the form
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.
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.
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.
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.
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
(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
The pole terms corresponding to the matrix elements of the on-shell physical states,
Let’s make a special gauge transformation from covariant gauge to Coulomb type gauge to obtain the momentum and angular momentum operators,
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
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.