# On the role of the subsidiary fields in the momentum and angular momentum in
covariantly quantized QED and QCD

Elliot Leader<EMAIL_ADDRESS>Blackett laboratory
Imperial College London
Prince Consort Road
London SW7 2AZ, UK

###### Abstract

The covariant quantization of QED and QCD requires the introduction of
subsidiary gauge-fixing and ghost fields and it is crucial to understand the
role of these in the energy-momentum tensor, and in the momentum and angular
momentum operators. These issues were studied in Leader (2011), and key
results from this study were utilized in the major review of angular momentum
in Leader and Lorce (2013). Damski Damski (2021) has rightly criticized as
incorrect the derivation of certain equations in Leader (2011). We show,
however, that the key results of Leader (2011) which are utilized in Leader
and Lorce (2013) are unaffected by Damski’s criticism.

###### pacs:

11.15.-q, 12.20.-m, 12.38.Aw, 12.38.Bx, 12.38.-t, 14.20.Dh

##  I  Introduction

It is well known that the covariant quantization of QED and QCD i.e. in which
the photon vector potential $A^{\mu}(x)$ and the gluon vector potential
$A^{\mu}_{a}$ transform as genuine Lorentz 4-vectors, is a non-trivial task
Lautrup (1967); Nakanishi (1966, 1972); Kugo and Ojima (1979) involving the
introduction of a scalar gauge-fixing field (Gf) in QED and both a gauge-
fixing field and Faddeev-Popov ghosts fields (Gf+Gh) in QCD. In both QED and
QCD the expressions for the linear and angular momentum _operators_ include
terms involving all these fields.
As explained in detail in Leader (2011) it is necessary to work in an
indefinite-metric space i.e. one in which the “length” or norm of a vector can
be either positive or negative and the definition of the “physical states”
with positive norm has to specified with care. It is also necessary to specify
the condition for an operator to represent a physically measurable quantity
i.e. to be an “observable”.
The situation is further complicated by the fact that it is possible to deal
with different versions of the energy-momentum tensor $t^{\mu\nu}$, of which
the two most important are the Canonical version $t^{\mu\nu}_{can}$ which
follows from Noether’s theorem and the Bellinfante version $t^{\mu\nu}_{bel}$
which is symmetric under $(\mu\leftrightarrow\nu)$ and which differs from
$t^{\mu\nu}_{can}$ by a divergence term, as will be spelled out in detail
below. Both these versions of the energy-momentum tensor are conserved
quantities. Based on these one can define, in the standard way, the momentum
operators $P^{\mu}_{can}$ and $P^{\mu}_{bel}$ and the angular momentum
operators $J^{i}_{can}$ and $J^{i}_{bel}$. Because the energy-momentum tensors
are related by a divergence one can show that for the matrix elements between
any normalizable physical states,
$\langle\Phi|\,P^{\mu}_{bel}\,|\Psi\rangle=\langle\Phi|\,P^{\mu}_{can}\,|\Psi\rangle$
and
$\langle\Phi|\,J^{i}_{bel}\,|\Psi\rangle=\langle\Phi|\,J^{i}_{can}\,|\Psi\rangle$.
Of key physical interest in QCD is the question of the fraction of the
momentum and angular momentum carried by quarks and gluons in a hadron, with
analogous questions about electrons and photons in QED. Clearly to answer
these questions one has to know the contributions of the subsidiary fields to
the physical matrix elements of the above operators. In all papers dealing
with the QCD case it is _assumed, without comment_ , that the contribution
from the subsidiary fields is zero. A proof of this was given in Leader
(2011), which was based on the demonstration that for physical matrix elements
in QED
$\langle\Phi^{\prime}|\,t^{\mu\nu}_{bel}(Gf)\,|\Phi\rangle=\langle\Phi^{\prime}|\,t^{\mu\nu}_{can}(Gf)\,|\Phi\rangle=0$
and for QCD
$\langle\Phi^{\prime}|\,t^{\mu\nu}_{bel}(Gf+Gh)\,|\Phi\rangle=\langle\Phi^{\prime}|\,t^{\mu\nu}_{can}(Gf+Gh)\,|\Phi\rangle=0$.
Damski Damski (2021) has pointed out that the proof for both the Canonical and
Bellinfante cases in QED is wrong, because it assumes that the physical states
alone form a complete set and thus uses $1=\sum|\Phi\rangle\langle\Phi|$,
which is an incorrect “resolution of the identity”, because it leaves out the
states of negative norm. Although he does not comment on QCD, Damski’s
argument shows that the proof that
$\langle\Phi^{\prime}|\,t^{\mu\nu}_{can}(Gf+Gh)\,|\Phi\rangle=0$ in Leader
(2011) is also incorrect.
This criticism seems to imply that the conclusion reached in Leader (2011)
that the subsidiary fields do not contribute to the momentum and angular
momentum is false, but as will be shown, this implication is wrong.
We shall show the following:
a) while, indeed, for the Canonical case

$\langle\Phi^{\prime}|\,t^{\mu\nu}_{can}(Gf)\,|\Phi\rangle\neq
0\quad\textrm{and}\quad\langle\Phi^{\prime}|\,t^{\mu\nu}_{can}(Gf+Gh)\,|\Phi\rangle\neq
0$ (1)

on the contrary, for the Bellinfante case, as claimed in Leader (2011)

$\langle\Phi^{\prime}|\,t^{\mu\nu}_{bel}(Gf)\,|\Phi\rangle=0\quad\textrm{and}\quad\langle\Phi^{\prime}|\,t^{\mu\nu}_{bel}(Gf+Gh)\,|\Phi\rangle=0.$
(2)

b) despite Eq. (1) one has

$\partial_{\mu}\,\langle\Phi^{\prime}|\,t^{\mu\nu}_{can}(Gf)\,|\Phi\rangle=\partial_{\mu}\,\langle\Phi^{\prime}|\,t^{\mu\nu}_{can}(Gf+Gh)\,|\Phi\rangle=0$
(3)

and obviously an analogous result for the Bellinfante case as a consequence of
(2), which are key results utilized in the major review Leader and Lorce
(2013) of the ”Angular Momentum Controversy”.
c) despite Eq, (1) it turns out that the subsidiary fields do not contribute
to the physical matrix elements of either the Canonical or Bellinfante
versions of the momentum or angular momentum. This crucial result, as
mentioned above, is normally assumed without comment in papers on QCD.

##  II Physical matrix elements of the Bellinfante energy-momentum tensor

We shall show, as claimed in Leader (2011), that the physical matrix elements
of the gauge-fixing and ghost contributions $t^{\mu\nu}_{bel}(Gf+Gh)$ in QCD
and the gauge-fixing contribution $t^{\mu\nu}_{bel}(Gf)$ in QED, vanish. The
proof given for the QED case in Leader (2011), as pointed out by Damski Damski
(2021), is incorrect, but the result is actually true . Surprisingly it turns
out that the QCD case is simpler to deal with than the QED case , which can be
derived as a special case.

###  II.1  Quantum Chromodynamics

The pure quark-gluon Lagrangian ${\cal{L}}_{qG}$ is

${\cal{L}}_{qG}=-\frac{1}{4}G^{a}_{\mu\nu}G^{\mu\nu}_{a}+\frac{1}{2}\bar{\psi}^{l}[\delta_{lm}\,i\,(\overrightarrow{\not\\!{\partial}}-\overleftarrow{\not\\!{\partial}})-2\,gt^{a}_{lm}\,\not{\negthickspace}{A}^{a}]\psi^{m}.$
(4)

In order to quantize the theory covariantly one has to introduce both a gauge-
fixing field $B(x)$ and Fadeev-Popov anti-commuting fermionic ghost fields
$c(x),\,\bar{c}(x)$. The Kugo-Ojima Lagrangian Kugo and Ojima (1978) for the
covariantly quantized theory is then

${\cal{L}}={\cal{L}}_{qG}+{\cal{L}}_{Gf+Gh}$ (5)

where

${\cal{L}}_{Gf+Gh}=-i(\partial^{\mu}\bar{c}^{a})D^{ab}_{\mu}c_{b}-(\partial^{\mu}B^{a})A^{a}_{\mu}+\frac{\textsf{a}}{2}B^{a}B^{a}.$
(6)

The physical states $|\Psi\rangle$ are defined by the subsidiary conditions

$Q_{B}|\Psi\rangle=0$ (7) $Q_{c}|\Psi\rangle=0$ (8)

where the conserved, hermitian charge $Q_{B}$ is given by

$Q_{B}=\int
d^{3}x[B^{a}{\overleftrightarrow{\partial}}_{0}c^{a}-gB^{a}f_{abc}A^{b}_{0}c^{c}-i(g/2)(\partial_{0}\bar{c}^{a})f_{abc}c^{b}c^{c}].$
(9)

and the conserved charge $Q_{c}$

$Q_{c}=\int
d^{3}x[\bar{c}^{a}\overleftrightarrow{\partial}_{0}c^{a}-g\bar{c}^{a}f_{abc}A^{b}_{0}c^{c}]$
(10)

“measures” the _ghost number_

$i[Q_{c},\phi]=N\phi$ (11)

where $N=1$ for
$\phi=c,\,\,-1\,\,\textrm{for}\,\,\phi=\bar{c}\,\,\textrm{and}\,\,0$ for all
other fields.

The Bellinfante energy-momentum tensor is

$t^{\mu\nu}_{bel}=t^{\mu\nu}_{bel}(qG)+t^{\mu\nu}_{bel}(Gf+Gh)$ (12)

where

$t^{\mu\nu}_{bel}(qG)=\frac{i}{4}[\bar{\psi}_{l}\gamma^{\mu}\overleftrightarrow{D}^{\nu}\psi_{l}+(\mu\leftrightarrow\nu)]-G^{\mu\beta}_{a}G^{\nu}_{a\beta}-g^{\mu\nu}{\cal{L}}_{qG}$
(13)

and the gauge-fixing and ghost terms are given by

$t^{\mu\nu}_{bel}(Gf+Gh)=-(A^{\mu}_{a}\partial^{\nu}B_{a}+A^{\nu}_{a}\partial^{\mu}B_{a})-i[(\partial^{\mu}\bar{c}_{a})D^{\nu}_{ab}c_{b}+(\partial^{\nu}\bar{c}_{a})D^{\mu}_{ab}c_{b}]-g^{\mu\nu}{\cal{L}}_{Gf+Gh}.$
(14)

This can be rewritten Kugo and Ojima (1979) as an anti-commutator with $Q_{B}$

$t^{\mu\nu}_{bel}(Gf+Gh)=-\\{Q_{B},\,\big{(}(\partial^{\mu}\bar{c}_{a})A^{\nu}_{a}+(\partial^{\nu}\bar{c}_{a})A^{\mu}_{a}+g^{\mu\nu}[\frac{\texttt{a}}{2}\bar{c}_{a}B_{a}-(\partial^{\rho}\bar{c}_{a})A_{\rho}^{a}]\big{)}\\}.$
(15)

It follows from Eqs. (7) and (15) that $t^{\mu\nu}_{bel}(Gf+Gh)$ does not
contribute to physical matrix elements i.e.

$\langle\Phi^{\prime}|\,t^{\mu\nu}_{bel}(Gf+Gh)\,|\Phi\rangle=0$ (16)

so that

$\langle\Phi^{\prime}|\,t^{\mu\nu}_{bel,QCD}\,|\Phi\rangle=\langle\Phi^{\prime}|\,t^{\mu\nu}_{bel}(qG)\,|\Phi\rangle$
(17)

and hence that the subsidiary fields do not contribute to the QCD expressions
for either $P^{\mu}_{bel}$ or $\bm{J}_{bel}$ i.e.

$\langle\Phi|\,P^{\mu}_{bel}\,|\Phi\rangle\equiv\int
d^{3}x\langle\Phi|\,t^{0\mu}_{bel}\,|\Phi\rangle=\int
d^{3}x\langle\Phi|\,t^{0\mu}_{bel}(qG)\,|\Phi\rangle=\langle\Phi|\,P^{\mu}_{bel}(qG)\,|\Phi\rangle$
(18)

and

$\displaystyle\langle\Phi|\,J^{i}_{bel}\,|\Phi\rangle\equiv\frac{1}{2}$
$\displaystyle\epsilon^{ijk}\ \int
d^{3}x\,\langle\Phi|\,x^{j}t^{0k}_{bel}\,-x^{k}t^{0j}_{bel}\,|\Phi\rangle$
$\displaystyle=\frac{1}{2}$ $\displaystyle\,\epsilon^{ijk}\,\int
d^{3}x\,\langle\Phi|\,x^{j}t^{0k}_{bel}(qG)\,-x^{k}t^{0j}_{bel}(qG)\,|\Phi\rangle=\langle\Phi|\,J^{i}_{bel}(qG)\,|\Phi\rangle.$
(19)

###  II.2  Quantum Electrodynamics

The most general covariantly quantized version of QED is given by the Lautrup-
Nakanishi Lagrangian density Lautrup (1967); Nakanishi (1966), which is a
combination of the Classical Lagrangian ($Clas$) and a Gauge Fixing part
($Gf$)

${\cal{L}}={\cal{L}}_{Clas}+{\cal{L}}_{Gf}$ (20)

where

${\cal{L}}_{Clas}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}\,[\bar{\psi}(i\not\\!{\partial}-m+e\not\negthickspace{A})\psi+\textrm{h.c.}]$
(21)

and

${\cal{L}}_{Gf}=-\partial_{\mu}B(x).A^{\mu}(x)+\frac{\textsf{a}}{2}B^{2}(x)$
(22)

where $B(x)$ is the gauge-fixing field and the parameter a determines the
structure of the photon propagator and is irrelevant for the present
discussion111The case $\textsf{a}=1$ corresponds to the Gupta-Bleuler approach
(see e.g. Jauch and Rohrlich (1955)) based on the Fermi Lagrangian. Note also
that in order to conform to the conventions used in the QCD case, the
expression for ${\cal{L}}_{Gf}$ differs from Nakanishi-Lautrup by a
4-divergence.

The _physical_ states $|\Phi\rangle$ of the theory are defined to satisfy

$B^{(+)}(x)|\Phi\rangle=0$ (23)

where

$B(x)=B^{(+)}(x)+B^{(-)}(x)$ (24)

with $B^{(\pm)}(x)$ the positive/negative frequency parts of $B(x)$.

For the conserved Bellinfante density one finds,

$t^{\mu\nu}_{bel}=\theta^{\mu\nu}_{bel}+t^{\mu\nu}_{bel}(Gf)$ (25)

where $\theta^{\mu\nu}_{bel}$, which is referred to as the classical energy
momentum tensor density, is

$\theta^{\mu\nu}_{bel}=\frac{i}{4}\,\bar{\psi}(\gamma^{\mu}\overleftrightarrow{D}^{\nu}+\gamma^{\nu}\overleftrightarrow{D}^{\mu})\,\psi-F^{\mu\beta}F^{\nu}_{\phantom{\nu}\beta}-g^{\mu\nu}{\cal{L}}_{Clas}$
(26)

where
$\overleftrightarrow{D}^{\nu}=\overleftrightarrow{\partial}^{\nu}-2ieA^{\nu}$,
and

$t^{\mu\nu}_{bel}(Gf)=-A^{\nu}\,\partial^{\mu}B-A^{\mu}\,\partial^{\nu}B-g^{\mu\nu}{\cal{L}}_{Gf}$
(27)

As explained in Kugo-Ojima Kugo and Ojima (1978) the QED expressions can be
obtained from the QCD case by putting the structure constants to zero and
suppressing the colour group labels, in whch case the gauge-fixing field $B$
and the ghost fields $c$ and $\bar{c}$ become free fields. The expression Eq.
(9) for the charge $Q_{B}$ then becomes, in terms of creation and annihilation
operators

$Q_{B}=i\int\frac{d^{3}k}{(2\pi)^{3}2E_{k}}[c_{k}^{\dagger}B_{k}-B_{k}^{\dagger}c_{k}].$
(28)

Because the Fadeev-Popov ghosts are here free, the state vector space
$\mathcal{V}$ can be decomposed into a direct product
$\mathcal{V}=\mathcal{V}_{phys}\otimes\mathcal{V}_{FP}$ where
$\mathcal{V}_{phys}$ is the usual QED physical state vector space. Moreover
the ghosts are redundant, so that one can work in the sector containing
neither $c$ nor $\bar{c}$ ghosts i.e.
$\mathcal{V}_{phys}\otimes|0\rangle_{FP}$. Hence the physical states in QED
can be taken to be $|\Phi\rangle\otimes|0\rangle_{FP}$. Using this and (28),
Eq. (7) can then be shown to imply (23). Hence

$\langle\Phi^{\prime}|\,t^{\mu\nu}_{bel,QED}(Gf)\,|\Phi\rangle=_{FP}\langle
0|\otimes\langle\Phi^{\prime}|\,t^{\mu\nu}_{bel,Free}(Gf+Gh)\,|\Phi\rangle\otimes|0\rangle_{FP}=0$
(29)

where $t^{\mu\nu}_{bel,Free}(Gf+Gh)$ is $t^{\mu\nu}_{bel,QCD}(Gf+Gh)$ in which
the structure constants are put to zero and the gauge-fixing and ghost fields
are free. Thus

$\langle\Phi^{\prime}|\,t^{\mu\nu}_{bel,QED}\,|\Phi\rangle=\langle\Phi^{\prime}|\,\theta^{\mu\nu}_{bel}\,|\Phi\rangle$
(30)

and hence, as in QCD, the subsidiary fields do not contribute to the QED
expressions for the physical matrix elements of either $P^{\mu}_{bel}$ or
$\bm{J}_{bel}$.

###  II.3  QED: direct study of subsidiary fields

The argument showing that the gauge fixing field in QED does not contribute to
the physical expectation value of the Bellinfante version of the energy-
momentum tensor and hence does not contribute to the expectation values of the
Bellinfante versions of the momentum and angular momentum, based on the QCD
case, is rather abstract, so we here show that the concrete expression for the
gauge fixing contribution to the Bellinfante angular momentum, given in
Damski’s paper Damski (2021), which deals only with the free electromagnetic
case, actually vanishes i.e. we shall show directly that

$\langle\Phi|\,\mathbf{J}_{bel}(\textrm{Gf})\,|\Phi\rangle=0$ (31)

where, in Damski’s notation

$J^{i}_{bel}(\textrm{Gf})=J^{i}_{\textrm{div}}+J^{i}_{\xi}.$ (32)

Consider

$\langle\Phi|J^{i}_{bel}(\textrm{Gf})|\Phi\rangle=\langle\Phi|J^{i}_{\textrm{div}}+J^{i}_{\xi}|\Phi\rangle=\int
d^{3}z\epsilon^{imn}\langle\Phi|z^{m}\mathcal{I}_{n}(z)-z^{n}\mathcal{I}_{m}(z)|{\Phi}\rangle$
(33)

with, following Damski,

$\mathcal{I}_{n}(z)=A_{n}\partial_{j}F_{0j}+(\partial\cdot
A)\partial_{n}A_{0}.$ (34)

Defining

$B(z)\equiv-\partial\cdot A$ (35)

and using the fact that in Damski the fields are free, one obtains

$\mathcal{I}_{n}(z)=A_{n}\partial_{0}B-B\partial_{n}A_{0}.$ (36)

We split the fields into their positive and negative frequency parts

$B(z)=B^{(+)}+B^{(-)}\qquad B^{(-)}=[B^{(+)}]^{\dagger}\quad\textrm{and}\quad
A_{\mu}(z)=A_{\mu}^{(+)}+A_{\mu}^{(-)}\qquad
A_{\mu}^{(-)}=[A_{\mu}^{(+)}]^{\dagger}$ (37)

with

$A_{\mu}^{(+)}(z)=\int[dk^{\prime}]\sum_{\sigma=0}^{\sigma=3}\epsilon_{\mu}(\textbf{k}^{\prime},\sigma)c_{\textbf{k}^{\prime}\sigma}e^{-ik^{\prime}\cdot
z}$ (38)

where the $\epsilon_{\mu}$ are polarization vectors and the
$c_{\textbf{k}^{\prime}\sigma}$ are annihilation operators and we use the
shorthand

$[dk^{\prime}]\equiv\frac{d^{3}k^{\prime}}{(2\pi)^{3/2}\sqrt{2\omega_{k^{\prime}}}}.$
(39)

In Damski’s notation one has

$B^{(+)}(z)=-i\int[dk]\omega_{k}L_{\mathbf{k}}\omega e^{-ik\cdot z}$ (40)

where

$L_{\mathbf{k}}=c_{\mathbf{k}3}-c_{\mathbf{k}0}.$ (41)

The physical states, as usual, are defined to satisfy

$B^{(+)}(z)|\Phi\rangle=0\qquad\langle\Phi|B^{(-)}(z)=0.$ (42)

Using these and the fact that the commutators
$[B^{(+)},A_{\mu}^{(+)}]=[B^{(-)},A_{\mu}^{(-)}]=0$, it follows that

$\displaystyle\langle\Phi|\mathcal{I}_{n}(z)|\Phi\rangle=\langle\Phi|A_{n}\partial_{0}B^{(-)}-B^{(+)}\partial_{n}A_{0}|\Phi\rangle$
$\displaystyle=\langle\Phi|[A^{(+)}_{n},\partial_{0}B^{(-)}]-[B^{(+)},\partial_{n}A^{(-)}_{0}]|\Phi\rangle$
$\displaystyle=\langle\Phi|\Phi\rangle\\{[A^{(+)}_{n},\partial_{0}B^{(-)}]-[B^{(+)},\partial_{n}A^{(-)}_{0}]\\}$
(43)

the last step following since the commutators are c-numbers.
The relevant commutators and polarization vectors are

$[L_{\bm{k}},c^{\dagger}_{\bm{k}^{\prime}0}]=-[c_{\bm{k}0},c^{\dagger}_{\bm{k}^{\prime}0}]=\delta^{3}(\bm{k}^{\prime}-\bm{k})\qquad\quad[L_{\bm{k}},c^{\dagger}_{\bm{k}^{\prime}3}]=[c_{\bm{k}3},c^{\dagger}_{\bm{k}^{\prime}3}]=\delta^{3}(\bm{k}^{\prime}-\bm{k})$
(44)

and

$\epsilon^{\mu}(\bm{k},0)=(1,\bm{0})\qquad\qquad\epsilon^{\mu}(\bm{k},3)=(0,\bm{k}/\omega_{k}).$
(45)

Substituting Eqs. (38, 40, 44, and 45) into Eq. (II.3) , we find, after some
labour, that

$[A^{(+)}_{n},\partial_{0}B^{(-)}]-[B^{(+)},\partial_{n}A^{(-)}_{0}]=0$ (46)

and thus that in QED

$\langle\Phi|J^{i}_{bel,QED}(\textrm{Gf})|\Phi\rangle=0$ (47)

in agreement with the more general derivation based on the QCD case.

###  II.4 Bellinfante summary

To summarize, despite the incorrect proof for the QED case given in Leader
(2011) , the physical matrix elements of the gauge-fixing and ghost
contributions to the Bellinfante version of the energy-momentum tensor, in
both QED and QCD, actually do vanish. Hence the only contributions to the
momentum $P^{\mu}_{bel}$ and angular momentum $\bm{J}_{bel}$ are from photons
and electrons in the QED case and from quarks and gluons in QCD.
This latter property was thus correctly utilized in the review paper Leader
and Lorce (2013). Also, in writing down the most general structure for the
matrix elements of $\langle\Phi^{\prime}|\,t^{\mu\nu}_{bel}(qG)\,|\Phi\rangle$
in Leader and Lorce (2013), use was made of the claim that

$\partial_{\mu}\,\langle\Phi^{\prime}|\,t^{\mu\nu}_{bel}(qG)\,|\Phi\rangle=0.$
(48)

This follows because the total $t^{\mu\nu}_{bel}$ is a conserved operator and,
via (17),

$\partial_{\mu}\,\langle\Phi^{\prime}|\,t^{\mu\nu}_{bel}(qG)\,|\Phi\rangle=\partial_{\mu}\,\langle\Phi^{\prime}|\,t^{\mu\nu}_{bel}\,|\Phi\rangle=0.$
(49)

Note also that this justifies the results in several papers in the literature,
e.g. Ji Ji (1997, 1997, 1998), Jaffe and Manohar Jaffe and Manohar (1990),
Bakker, Leader and Trueman (BLT) Bakker  _et al._ (2004) and Wakamatsu
Wakamatsu (2010, 2011), where the general structure of the physical matrix
elements of $t^{\mu\nu}_{bel}(qG)$ (or its QED analogue) is derived under the
_unstated assumption_ that Eq. (49) holds.

##  III Physical matrix elements of the Canonical energy-momentum tensor

In Leader (2011) it was claimed that the physical matrix elements of the
gauge-fixing and ghost contributions $t^{\mu\nu}_{can}(Gf+Gh)$ in QCD and the
gauge-fixing contribution $t^{\mu\nu}_{can}(Gf)$ in QED, vanish. The result
given for the QED case in Leader (2011), as pointed out by Damski Damski
(2021), is wrong because of an incorrect use of the ”resolution of the
identity” for a space with an indefinite metric, and, although not discussed
by Damski, also the QCD result is wrong for the same reason.
Thus, in contrast to the Bellinfante case and contrary to the claims made in
Leader (2011), in QCD

$\langle\Phi^{\prime}|\,t^{\mu\nu}_{can}(Gf+Gh)\,|\Phi\rangle\neq 0$ (50)

and in QED

$\langle\Phi^{\prime}|\,t^{\mu\nu}_{can}(Gf)\,|\Phi\rangle\neq 0.$ (51)

We shall analyze the consequences of these for the QCD case. Completely
analogous arguments hold for the case of QED.
There are two questions which have to be answered. Given (50):
1) Is the analysis of the general structure of the physical matrix elements of
the quark-gluon $t^{\mu\nu}_{can}(qG)$ given in Leader and Lorce (2013)
correct?
2) Do the subsidiary fields contribute to the physical matrix elements of the
Canonical momentum $P^{\mu}_{can}$ and angular momentum $\bm{J}_{can}$ ?

###  III.1  Structure of the physical matrix elements of the quark-gluon
$t^{\mu\nu}_{can}(qG)$

The Bellinfante and Canonical energy-momentum tensors differ from each other
by a divergence term of the following form:

$t^{\mu\nu}_{can}=t^{\mu\nu}_{bel}-\partial_{\lambda}G^{\lambda\mu\nu},$ (52)

where the so-called _superpotential_ reads

$G^{\lambda\mu\nu}=\tfrac{1}{2}\left(M^{\lambda\mu\nu}_{\text{spin}}+M^{\mu\nu\lambda}_{\text{spin}}+M^{\nu\mu\lambda}_{\text{spin}}\right),$
(53)

and, crucially, is antisymmetric w.r.t. its first two indices

$G^{\lambda\mu\nu}=-G^{\mu\lambda\nu}.$ (54)

The $M_{spin}$ involves a sum over all fields and is given in terms of the
Lagrangian by

$M_{\text{spin}}^{\mu\nu\rho}(x)=-i\sum_{allfields}\,\frac{\partial\cal{L}}{\partial(\partial_{\mu}\phi_{r})}\,(\Sigma^{\nu\rho})^{\phantom{r}s}_{r}\phi_{s}(x).$
(55)

where
$(\Sigma^{\mu\nu})^{\phantom{r}s}_{r}=-(\Sigma^{\nu\mu})^{\phantom{r}s}_{r}$
is an operator related to the spin of the field. For example, for particles
with the most common spins, one has

spin-$0$ particle $\displaystyle\phi(x)$
$\displaystyle(\Sigma^{\mu\nu})^{\phantom{r}s}_{r}$ $\displaystyle=0,$ (56)
spin-$1/2$ Dirac particle $\displaystyle\psi_{r}(x)$
$\displaystyle(\Sigma^{\mu\nu})^{\phantom{r}s}_{r}$
$\displaystyle=\tfrac{1}{2}\left(\sigma^{\mu\nu}\right)^{\phantom{r}s}_{r},$
(57) spin-$1$ particle $\displaystyle A_{\alpha}(x)$
$\displaystyle(\Sigma^{\mu\nu})^{\phantom{\alpha}\beta}_{\alpha}$
$\displaystyle=i\left(\delta^{\mu}_{\alpha}\,g^{\nu\beta}-\delta^{\nu}_{\alpha}\,g^{\mu\beta}\right).$
(58)

Examination of the structure of the Lagrangians in Eqs. (4, 6) shows that
since ${\cal{L}}_{qG}$ does not contain any gauge-fixing or ghost fields and
${\cal{L}}_{Gf+Gh}$ does not contain any derivatives of $A^{\mu}_{a}$ we may
write

$M_{\text{spin}}^{\mu\nu\rho}(x)=M_{\text{spin}}^{\mu\nu\rho}(x)|_{qG}+M_{\text{spin}}^{\mu\nu\rho}(x)|_{Gf+Gh}$
(59)

and thus in Eq. (52)

$G^{\lambda\mu\nu}=G^{\lambda\mu\nu}|_{qG}+G^{\lambda\mu\nu}|_{Gf+Gh}$ (60)

so that separately

$t^{\mu\nu}_{can}(qG)=t^{\mu\nu}_{bel}(qG)-\partial_{\lambda}G^{\lambda\mu\nu}|_{qG}$
(61)

and

$t^{\mu\nu}_{can}(Gf+Gh)=t^{\mu\nu}_{bel}(Gf+Gh)-\partial_{\lambda}G^{\lambda\mu\nu}|_{Gf+Gh}.$
(62)

Hence,

$\displaystyle\partial_{\mu}\,t^{\mu\nu}_{can}(Gf+Gh)$ $\displaystyle=$
$\displaystyle\partial_{\mu}\,t^{\mu\nu}_{bel}(Gf+Gh)-\partial_{\mu}\partial_{\lambda}G^{\lambda\mu\nu}|_{Gf+Gh}$
(63) $\displaystyle=$
$\displaystyle\partial_{\mu}\,t^{\mu\nu}_{bel}(Gf+Gh)\quad\textrm{via}\,\,(\ref{Gsym})$

and thus for the physical matrix elements, using (16),

$\partial_{\mu}\,\langle\Phi^{\prime}|\,t^{\mu\nu}_{can}(Gf+Gh)\,|\Phi\rangle=\partial_{\mu}\,\langle\Phi^{\prime}|\,t^{\mu\nu}_{bel}(Gf+Gh)\,|\Phi\rangle=0.$
(64)

Finally, then, since the total $t^{\mu\nu}_{can}$ is a conserved operator, we
obtain the key result

$\partial_{\mu}\,\langle\Phi^{\prime}|\,t^{\mu\nu}_{can}(qG)\,|\Phi\rangle=\partial_{\mu}\,\langle\Phi^{\prime}|\,t^{\mu\nu}_{can}\,|\Phi\rangle=0$
(65)

and the analysis of the general structure of the physical matrix elements of
the quark-gluon $t^{\mu\nu}_{can}(qG)$ given in Leader and Lorce (2013), which
relied on this property, is correct.

###  III.2 Contribution of the subsidiary fields to the physical matrix
elements of the Canonical momentum $P^{\mu}_{can}$ and angular momentum
$\bm{J}_{can}$

From Eqs. (18) and (61)

$\displaystyle\langle\Phi|\,P^{\mu}_{bel}\,|\Phi\rangle$ $\displaystyle=$
$\displaystyle\int d^{3}x\langle\Phi|\,t^{0\mu}_{bel}(qG)\,|\Phi\rangle$ (66)
$\displaystyle=$ $\displaystyle\int
d^{3}x\langle\Phi|\,t^{0\mu}_{can}(qG)\,|\Phi\rangle+\int
d^{3}x\langle\Phi|\,\partial_{\lambda}G^{\lambda 0\nu}|_{qG}\,|\Phi\rangle.$

By the antisymmetry property (54) the last term is actually a 3-dimensional
divergence $\partial_{i}G^{i0\nu}|_{qG}$, yielding a surface term at infinity,
which, as always, is assumed to vanish. Thus

$\langle\Phi|\,P^{\mu}_{bel}\,|\Phi\rangle=\int
d^{3}x\langle\Phi|\,t^{0\mu}_{can}(qG)\,|\Phi\rangle=\langle\Phi|\,P^{\mu}_{can}(qG)\,|\Phi\rangle.$
(67)

But

$\langle\Phi|\,P^{\mu}_{bel}\,|\Phi\rangle=\langle\Phi|\,P^{\mu}_{can}\,|\Phi\rangle$
(68)

so that, indeed,

$\langle\Phi|\,P^{\mu}_{can}\,|\Phi\rangle=\langle\Phi|\,P^{\mu}_{can}(qG)\,|\Phi\rangle$
(69)

and the subsidiary fields do not contribute to the physical matrix elements of
$P^{\mu}_{can}$. A similar argument shows that

$\langle\Phi|\,J^{i}_{can}\,|\Phi\rangle=\langle\Phi|\,J^{i}_{can}(qG)\,|\Phi\rangle$
(70)

and the subsidiary fields also do not contribute to the physical matrix
elements of $J^{i}_{can}$.

###  III.3 Canonical summary

To summarize, despite the incorrect proof given in Leader (2011) concerning
the physical matrix elements of the gauge-fixing contributions in QED and the
gauge-fixing and ghost contributions in QCD to the Canonical version of the
energy-momentum tensor, the essential property used in writing down the most
general structure for the QCD matrix elements
$\langle\Phi^{\prime}|\,t^{\mu\nu}_{can}(qG)\,|\Phi\rangle$ and the QED matrix
elements $\langle\Phi^{\prime}|\,\Theta^{\mu\nu}_{can}\,|\Phi\rangle$ in
Leader and Lorce (2013), namely, that

$\partial_{\mu}\,\langle\Phi^{\prime}|\,t^{\mu\nu}_{can}(qG)\,|\Phi\rangle=0\quad\textrm{and}\quad\partial_{\mu}\,\langle\Phi^{\prime}|\,\Theta^{\mu\nu}_{can}\,|\Phi\rangle=0$
(71)

is correct.
Moreover, despite the incorrect derivation in Leader (2011), the only
contributions to the physical matrix elements of the momentum $P^{\mu}_{can}$
and angular momentum $\bm{J}_{can}$ are from photons and electrons in the QED
case and from quarks and gluons in QCD. This latter property was thus
correctly utilized in the review paper Leader and Lorce (2013).

##  IV Conclusions

Damski’s criticism Damski (2021) of the _proof_ given in Leader (2011) for
certain properties of the physical matrix elements of the energy-momentum
tensor $t^{\mu\nu}$ in QED is valid, and it is also applicable to QCD, so
that, contrary to the assertions in Leader (2011), for the Canonical case

$\langle\Phi^{\prime}|\,t^{\mu\nu}_{can}(Gf)\,|\Phi\rangle\neq
0\quad\emph{and}\quad\langle\Phi^{\prime}|\,t^{\mu\nu}_{can}(Gf+Gh)\,|\Phi\rangle\neq
0.$ (72)

Nonetheless the following crucial features for QED and QCD, utilized in the
angular momentum review Leader and Lorce (2013), are in fact correct:
a) for the Bellinfante case, as claimed in Leader (2011)

$\langle\Phi^{\prime}|\,t^{\mu\nu}_{bel}(Gf)\,|\Phi\rangle=0\quad\emph{and}\quad\langle\Phi^{\prime}|\,t^{\mu\nu}_{bel}(Gf+Gh)\,|\Phi\rangle=0.$
(73)

b) despite Eq. (72) one has for the Canonical case,

$\partial_{\mu}\,\langle\Phi^{\prime}|\,t^{\mu\nu}_{can}(Gf)\,|\Phi\rangle=0\quad\emph{and}\quad\partial_{\mu}\,\langle\Phi^{\prime}|\,t^{\mu\nu}_{can}(Gf+Gh)\,|\Phi\rangle=0.$
(74)

and obviously an analogous result for the Bellinfante case as a consequence of
(73).
c) despite Eq. (72) the subsidiary fields do not contribute to the physical
matrix elements of either the Canonical or Bellinfante versions of the
momentum or angular momentum, a result which is normally _assumed without
comment_ in papers on QCD.

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