arXiv subjects
Publications and source records attributed to Wei-Min Sun.
We reply to the Comment of X. Ji [arXiv:0810.4913] on our paper [PRL 100:232002 (2008)], concerning angular momentum algebra, locality, Lorentz covariance, and measurability of our gauge-invariant description of the spin and orbital angular momentum of quarks and gluons.
Following our recent finding that the renowned formula $\vec x\times (\vec E\times\vec B)$ is not the correct density for the electromagnetic angular momentum, here we examine the validity of the Poynting vector $\vec E \times\vec B$ as the electromagnetic momentum density (or energy flux). The competitor is the gauge-invariant canonical momentum $E^i\vec \nabla A^i_\perp$. It often gives the same result as $\vec E\times\vec B$, but we propose that a delicate measurement (of the {\em azimuthal} energy flow in polarized atomic radiations) can make a discrimination. By clarifying the profound difference between two kinds of energy-momentum tensors: the canonical (or mechanical) one and the symmetric (or gravitational) one, we predict that it is $E^i\vec \nabla A^i_\perp$ that would pass the delicate experimental test. Our observations have far-reaching implications for understanding the source of gravity, and the nucleon momentum as well.
Parallel to the construction of gauge invariant spin and orbital angular momentum for QED in paper (I) of this series, we present here an analogous but non-trivial solution for QCD. Explicitly gauge invariant spin and orbital angular momentum operators of quarks and gluons are obtained. This was previously thought to be an impossible task, and opens a more promising avenue towards the understanding of the nucleon spin structure.
By differentiating the dressed quark propagator with respect to a variable background field, the linear response of the dressed quark propagator in the presence of the background field can be obtained. From this general method, using the vector background field as an illustration, we derive a general formula for the four-quark condensate $<{\tilde 0}|:{\bar q}(0)γ_μq(0){\bar q}(0)γ_μq(0):|{\tilde 0}>$. This formula contains the corresponding fully dressed vector vertex and it is shown that factorization for $<{\tilde 0}|:{\bar q}(0)γ_μq(0){\bar q}(0)γ_μq(0):| {\tilde 0}>$ holds only when the dressed vertex is taken to be the bare one. This property also holds for all other type of four-quark condensate.
The conflict between cononical commutation relation and gauge invariance, which both the momentum and angular momentum of quark and gluon should satisfy, is clarified. The quantum version of gauge invariance is studied. The gauge independence of the matrix elements of quark momentum and angular momentum operators between physical states are proved. We suggest to use the canonical quark momentum and angular momentum distributions to describe the nucleon internal structure in order to establish an internal consistent description of hadron spectroscopy and hadron structure. The same problem for the atomic spectroscopy and structure is discussed.
In this letter it is shown on general ground that there exist two qualitatively distinct solutions of the Dyson-Schwinger equation for the quark propagator in the case of non-zero current quark mass. One solution corresponds to the ``Nambu-Goldstone'' phase and the other one corresponds to the ``Wigner'' phase in the chiral limit.
The possible anomaly of the tensor current divergence equation in U(1) gauge theories is calculated by means of perturbative method. It is found that the tensor current divergence equation is free of anomalies.
The transverse axial vector and vector anomalies in four-dimensional U(1) gauge theories studied in [10] is reexamined by means of perturbative methods. The absence of transverse anomalies for both axial vector and vector current is verified. We also show that the Pauli-Villars regularization and dimensional regularization give the same result on the transverse anomaly of both axial vector and vector current.
In this paper we investigated the problem of the existence of invariant meaures on the local gauge group. We prove that it is impossible to define a {\it finite} translationally invariant measure on the local gauge group $C^{\infty}({\bf R}^n,G)$(where $G$ is an arbitrary matrix Lie group).
We give a simple proof that there does not exist a Haar measure on the group $C^{\infty}({\bf R}^n,U(1))$.
We evaluated some particular type of functional integral over the local gauge group C^{\infty}({\bf R}^n, U(1)) by going to a discretized lattice. The results explicitly violates the property of the Haar measure. We also analysed the Faddeev-Popov method through a toy example. The results also violates the property of the Haar measure.
We show that in the functional integral formalism of U(1) gauge field theory some formal manipulation such as interchange of order of integration can yield erroneous results. The example studied is analysed by Fubini theorem.
We prove that a kind of averaging procedure for constructing gauge-invariant operators(or functionals) out of gauge-variant ones is erroneous and inapplicable for a large class of operators(or functionals).
We find that sometimes the usual definition of functional integration over the gauge group through limiting process may have internal difficulties.
Based on a canonically derived path integral formalism, we demonstrate that the perturbative calculation of the matrix element for gauge dependent operators has crucial difference from that for gauge invariant ones. For a gauge dependent operator ${\cal O}(ϕ)$ what appears in the Feynman diagrams is not ${\cal O} (ϕ)$ itself, but the gauge-transformed one ${\cal O}(^ωϕ)$, where $ω$ characterizes the specific gauge transformation which brings any field variable into the particular gauge which we have adopted to quantize the gauge theory using the canonical method. The study of the matrix element of gauge dependent operators also reveals that the formal path integral formalism for gauge theory is not always reliable.