# A product picture for quantum electrodynamics

Bernard S. Kay∗
_Department of Mathematics, University of York, York YO10 5DD, UK_
<EMAIL_ADDRESS>
###### Abstract

We present a short account of our work to provide quantum electrodynamics with
a _product picture_. It aims to complement the longer exposition in a recent
paper in _Foundations of Physics_ and to help to make that work more
accessible. The product picture is a formulation of QED, equivalent to
standard Coulomb gauge QED, but in which the Hilbert space arises as (a
certain physical subspace of) a product of a Hilbert space for the
electromagnetic field and a Hilbert space for charged matter (i.e. the Dirac
field) and in which the Hamiltonian arises as the sum of an electromagnetic
Hamiltonian and a charged matter Hamiltonian and an interaction term. (The
Coulomb gauge formulation of QED is not a product picture because, in it, the
longitudinal part of the electromagnetic field is made out of charged matter
operators.) We also recall a ‘Contradictory Commutator Theorem’ for QED which
exposes flaws in previous attempts at temporal gauge quantization of QED and
we explain how our product picture appears to offer a way to overcome those
flaws. Additionally, we discuss the extent to which that theorem may be
generalized to Yang-Mills fields. We also develop a product picture for
nonrelativistic charged particles in interaction with the electromagnetic
field and point out how this leads to a novel way of thinking about the theory
of many nonrelativistic electrically charged particles with Coulomb
interactions. In an afterword, we explain how the provision of a product
picture for QED gives hope that one will be able likewise to have a product
picture for (Yang Mills and for) quantum gravity – the latter being needed to
make sense of the author’s _matter-gravity entanglement hypothesis_. Also, we
briefly discuss some similarities and differences between that hypothesis and
its predictions and ideas of Roger Penrose related to a possible rôle of
gravity in quantum state reduction and related to cosmological entropy.

##  1 Introduction: Coulomb Gauge QED and its aesthetically unpleasant
features

Let me begin by recalling the standard Coulomb gauge formulation for a charged
Dirac field, $\psi$, in interaction with the quantum electromagnetic field.
The Hamiltonian is

$H_{\mathrm{QED}}^{\mathrm{Coulomb}}=\int\frac{1}{2}{\bm{\pi}^{\perp}}^{2}\\!+\frac{1}{2}({\bm{\nabla}}{\bm{\times}}{\bm{A}})^{2}+\psi^{*}\gamma^{0}{\bm{\gamma}}{\bm{\cdot}}(-i{\bm{\nabla}}-{\rm
e}{\bm{A}})\psi+m\psi^{*}\gamma^{0}\psi\,d^{3}x+V_{\mathrm{Coulomb}}$ (1)

where

$V_{\mathrm{Coulomb}}=\frac{\rm
e^{2}}{2}\int\int\frac{\psi^{*}(\bm{x})\psi(\bm{x})\psi^{*}(\bm{y})\psi(\bm{y})}{4\pi|\bm{x}-\bm{y}|}\,d^{3}xd^{3}y\
$

is the standard Coulomb action-at-a-distance electrostatic potential energy of
the Dirac charge density

$\rho(\bm{x})={\rm e}\psi^{*}(\bm{x})\psi(\bm{x}).$ (2)

Here, $\rm e$ is the charge on the electron, $\bm{A}$ is the vector potential
for the magnetic field satisfying the Coulomb gauge condition,
$\bm{\nabla}\bm{\cdot A}=0$, and $\bm{\pi}^{\perp}$ is (minus) the transverse
part (satisfying $\bm{\nabla}\bm{\cdot\bm{\pi}^{\perp}}=0$) of the electric
field.

This is to be supplemented with the commutation and anticommutation relations

$[A_{i}(\bm{x}),{\pi^{\perp}}_{j}(\bm{y})]=i\delta_{ij}\delta^{(3)}(\bm{x}-\bm{y})+i\frac{\partial^{2}}{\partial
x^{i}\partial x^{j}}\left({1\over 4\pi|\bm{x}-\bm{y}|}\right),$ (3)
$\\{\psi(\bm{x}),\psi^{*}(\bm{y})\\}=\delta^{(3)}({\bm{x}}-{\bm{y}})$

while the commutators of $A_{i}(\bm{x})$ with $A_{j}(\bm{y})$,
$\pi^{\perp}_{i}(\bm{x})$ with $\pi^{\perp}_{j}(\bm{y})$, and of
$A_{i}(\bm{x})$ and $\pi^{\perp}_{i}(\bm{x})$ with $\psi(\bm{y})$ and
$\psi^{*}(\bm{y})$ vanish, and also the anticommutator of $\psi(\bm{x})$ with
$\psi(\bm{y})$ and of $\psi^{*}(\bm{x})$ with $\psi^{*}(\bm{y})$ vanish.

While it leaves unaddressed many questions of mathematical rigour (and the
related difficulties of renormalization) the Hamiltonian (1) together with the
commutation and anticommutation relations (3) provide us with a suitable
starting definition for what we know to be the correct theory of quantum
electrodynamics (QED). For example, QED is introduced in this way in Steven
Weinberg’s textbook on quantum field theory [12].

However, and leaving aside questions of mathematical rigour, this Coulomb-
gauge version of quantum electrodynamics suffers from a number of
aesthetically unpleasant features. An obvious such unpleasant feature is that
it makes implicit reference to a particular choice of Lorentz frame. (The way
our product picture transforms under Lorentz transformations is an interesting
question but we won’t address it here.) But, leaving that aside too, we wish
here to focus on the following:

  1. 1.

$V_{\mathrm{Coulomb}}$ is quartic in $\psi$ and also nonlocal.

  2. 2.

The commutation relations (3) also have a nonlocal piece in addition to the
delta function on the right hand side. (One can see that such a term is
necessary to be consistent with the transversality of ${\bm{\pi}}^{\perp}$ and
of the Coulomb gauge condition on $\bm{A}$ on the left hand side.)

  3. 3.

Only the transverse part of the electric field participates in the dynamics.
Instead of a longitudinal part, we have an action-at-a-distance force due to
the potential $V_{\mathrm{Coulomb}}$. This is unsatisfactory because, for
example, it suggests that we need to think of the energy stored in a capacitor
as due to the work required to separate its charged plates, or the work done
against the electrostatic attraction of opposite charges to charge them up,
whereas we would prefer (since Faraday!) to think it is stored in the
(longitudinal) electric field between the plates!

  4. 4.

True we can define the longitudinal electric field by
$\bm{E}^{\mathrm{long}}=-\bm{\nabla}\phi$ where

$\phi({\bm{x}})={\rm
e}\int\frac{\psi^{*}({\bm{y}})\psi({\bm{y}})}{4\pi|{\bm{x}}-{\bm{y}}|}\,d^{3}{\bm{y}}$
(4)

which, by the way, is equivalent to imposing Gauss’s law,
$\bm{\nabla.E}=\rho$, by definition! But then $E^{\mathrm{long}}$ is made out
of Dirac field operators and doesn’t have an independent existence (as well as
not participating in the dynamics).

  5. 5.

It would be nice if the Hilbert space, ${\cal H}_{\mathrm{QED}}$, could be
written as the tensor product,

${\cal H}_{\mathrm{QED}}={\cal H}_{\mathrm{electromag}}\otimes{\cal
H}_{\mathrm{charged\ matter}},$ (5)

of a Hilbert space, ${\cal H}_{\mathrm{electromag}}$, for the electromagnetic
field and a Hilbert space, ${\cal H}_{\mathrm{charged\ matter}}$, for charged
matter (i.e. for our Dirac field) and if also the QED Hamiltonian then arose
in the form

$H_{\mathrm{QED}}=H_{\mathrm{electromag}}+H_{\mathrm{charged\
matter}}+H_{\mathrm{interaction}}$ (6)

When such a statement is true of a quantum theory involving two systems in
interaction, we say it has a _product structure_. But the Coulomb-gauge
formulation of QED doesn’t have a “product structure” in this sense. It might
appear to do so if $H_{\mathrm{electromag}}$ were interpreted as the Hilbert
space of the transverse part of the electromagnetic field. But if we insist on
all electromagnetic field operators (both transverse and longitudinal) acting
on ${\cal H}_{\mathrm{electromag}}\otimes{\cal H}_{\mathrm{charged\ matter}}$
as operators of the form $A\otimes I$, where $I$ denotes the identity on on
${\cal H}_{\mathrm{charged\ matter}}$ – which is what we really mean when we
say that ${\cal H}_{\mathrm{electromag}}$ is “the Hilbert space for the
electromagnetic field” – then this clearly doesn’t hold since the longitudinal
part of the electric field is made out of charged matter field operators.

To amplify on Point 5, what would be preferable (and as we shall discuss
further in Section 5 not only for aesthetic reasons but also because it
enables one to ask and answer new physical questions) would be if there were a
way to quantize electrodynamics in which the Hilbert space took the product
form (5) and in which the Hamiltonian took the _product picture_ form (6), or
more precisely, if it took the form

$H^{\mathrm{PP}}_{\mathrm{QED}}=\int{1\over
2}{\tilde{\bm{\pi}}}^{2}\\!+{1\over
2}({\bm{\nabla}}{\bm{\times}}{\hat{\bm{A}}})^{2}+\psi^{*}\gamma^{0}{\bm{\gamma}}{\bm{\cdot}}(-i{\bm{\nabla}}-{\rm
e}{\hat{\bm{A}}})\psi+m\psi^{*}\gamma^{0}\psi\,d^{3}x$ (7)

with no Coulomb potential term. And if the electromagnetic field commutation
relations took the form

$[\hat{A}_{i}(\bm{x}),\tilde{\pi}_{j}(\bm{y})]=i\delta_{ij}\delta^{(3)}(\bm{x}-\bm{y}),\quad\\{\psi(\bm{x}),\psi^{*}(\bm{y})\\}=\delta^{(3)}(\bm{x}-\bm{y})$
(8)

with no nonlocal term. And if $\tilde{\bm{\pi}}$ was (minus) the _full_
electric field operator (rather than just its transverse part). And if Gauss’s
law, $\bm{\nabla.E}=\rho$, held as a genuine operator equation (rather than by
definition). And yet if this new formulation was entirely equivalent to
Coulomb gauge QED. Because after all, while we have complained that it is
aesthetically unpleasant, we do know that it’s physically correct!

I want next to show that (at a similarly mathematically non-rigorous level to
that at which the Coulomb gauge theory is typically discussed) it is possible
to have a formulation of QED, which we call the _product picture_ , equivalent
to the Coulomb gauge formulation, and with all these nice features. More
precisely, the equivalence is between the Coulomb gauge formulation and our
product picture theory restricted to a certain _product picture physical
subspace_ of a full (unphysical) product picture Hilbert space which we call
below the ‘augmented QED Hilbert space’.

As a preliminary to that, we conclude this introduction by recalling how the
operators $\bm{A}$ and $\bm{\pi}^{\perp}$ in Coulomb gauge QED may be
represented on a Hilbert space. In terms of the usual (say, momentum-space)
annihilation and creation operators, $a_{i}(\bm{k})$, $a_{i}^{+}(\bm{k})$ on
the Fock space, ${\cal F}({\cal H}_{\mathrm{one}})$ over the one particle
Hilbert space, ${\cal H}_{\mathrm{one}}=L^{2}({\mathbb{R}}^{3})^{3}$, we may
define

$A_{i}(\bm{k})=\left(\frac{a^{\mathrm{trans}}_{i}(\bm{k})+{a_{i}^{+}}^{\mathrm{trans}}(\bm{k})}{\sqrt{2}|k|^{1/2}}\right),\
{\pi^{\perp}}_{i}(\bm{k})=-i|k|^{1/2}\left(\frac{{a^{\mathrm{trans}}_{i}(\bm{k})-{a_{i}^{+}}^{\mathrm{trans}}}(\bm{k})}{\sqrt{2}}\right)$
(9)

where $a^{\mathrm{trans}}_{i}(\bm{k})$ is the transverse part,
$(\delta_{i}^{j}-k_{i}k^{j}/k^{2})a_{j}(\bm{k})$ of $a_{i}(\bm{k})$ and
${a^{+}_{i}}^{\mathrm{trans}}(\bm{k})$ the transverse part,
$(\delta_{i}^{j}-k_{i}k^{j}/k^{2})a^{+}_{j}(\bm{k})$ of $a_{i}^{+}(\bm{k})$.

It will be important to note here that we can think about the Hilbert space on
which $A_{i}(\bm{k})$ and ${\pi^{\perp}}_{i}(\bm{k})$ act in two slightly
different ways. Either we can think of it as the Fock space, ${\cal F}({\cal
H}_{\mathrm{one}}^{\mathrm{trans}})$, over the subspace, ${\cal
H}_{\mathrm{one}}^{\mathrm{trans}}$, consisting of transverse elements of
${\cal H}_{\mathrm{one}}$ or we can note that the full one particle Hilbert
space ${\cal H}_{\mathrm{one}}$ can be written as a sum of its two orthogonal
subspaces, ${\cal H}_{\mathrm{one}}^{\mathrm{trans}}$, and ${\cal
H}_{\mathrm{one}}^{\mathrm{long}}$ consisting respectively of transverse and
longitudinal vectors, and thus the full Fock space, ${\cal F}({\cal
H}_{\mathrm{one}})$, is the tensor product, ${\cal F}({\cal
H}_{\mathrm{one}}^{\mathrm{trans}})\otimes{\cal F}({\cal
H}_{\mathrm{one}}^{\mathrm{long}})$, and think of the subspace on which
$A_{i}(\bm{k})$ and ${\pi^{\perp}}_{i}(\bm{k})$ act as consisting of the
subspace ${\cal F}({\cal
H}_{\mathrm{one}}^{\mathrm{trans}})\otimes\Omega^{\mathrm{long}}$, where
$\Omega^{\mathrm{long}}$ is the vacuum vector in ${\cal
H}_{\mathrm{one}}^{\mathrm{long}}$. With the latter point of view, we admit
the possibility of _longitudinal photons_ but keep them in their vacuum state.

With the latter viewpoint, we can think of the full representation space of
QED (now incorporating the Dirac field too) as what we shall call the _Coulomb
gauge physical subspace (CGPS)_ , ${\cal F}({\cal
H}_{\mathrm{one}}^{\mathrm{trans}})\otimes\Omega^{\mathrm{long}}\otimes{\cal
H}_{\mathrm{Dirac}}$, of the _augmented QED Hilbert space_ , which is the name
we shall give to ${\cal F}({\cal H}_{\mathrm{one}})\otimes{\cal
H}_{\mathrm{Dirac}}$.

##  2  A product picture for QED

To provide QED with a product picture, the first step is to define operators,
$\hat{\bm{A}}$ and $\tilde{\bm{\pi}}$, on our augmented QED Hilbert space
${\cal F}({\cal H}_{\mathrm{one}})\otimes{\cal H}_{\mathrm{Dirac}}$ by

$\hat{A}_{i}(\bm{k})=\left(\frac{a_{i}(\bm{k})+{a_{i}^{+}}(\bm{k})}{\sqrt{2}|k|^{1/2}}\right),\quad\tilde{\bm{\pi}}=\bm{\pi}^{\perp}+\tilde{\bm{\pi}}^{\mathrm{long}}$

where $\bm{\pi}^{\perp}$ is defined as in Equation (9) and

$\tilde{\pi}^{\mathrm{long}}_{i}(\bm{k})=-\frac{2i|k|^{1/2}}{\sqrt{2}}a_{i}^{\mathrm{long}}(\bm{k}).$

This last definition may seem strange. Let us pause to notice that we could
think of $\tilde{\bm{\pi}}^{\mathrm{long}}$ as the result of first defining an
operator that we shall call $\hat{\bm{\pi}}$ by again taking its transverse
part to be the same as the definition of $\bm{\pi}^{\perp}$ in Equation (9)
and defining its longitudinal part, along similar lines, i.e. by

$\hat{\pi}^{\mathrm{long}}_{i}(\bm{k})=-i|k|^{1/2}\left(\frac{{a^{\mathrm{long}}_{i}(\bm{k})-{a_{i}^{+}}^{\mathrm{long}}}(\bm{k})}{\sqrt{2}}\right),$

but then deleting the second term on the right hand side containing a creation
operator and doubling the term on the right hand side containing an
annihilation operator. Clearly, in consequence (and unlike $\hat{\bm{A}}$ and
$\hat{\bm{\pi}}$) $\tilde{\bm{\pi}}$ will fail to be (even formally) self-
adjoint. However it will still have the same commutation relations with all
other operators (and in particular with $\hat{\bm{A}}$) that $\hat{\bm{\pi}}$
has and we shall soon see (see Theorem 1a below) that it _will_ be (formally)
self-adjoint on the appropriate ‘product picture physical subspace’ of the
augmented QED Hilbert space which we next define.

First we define the (formally) unitary operator

$U=\exp\left(i\int\hat{A}^{i}(\bm{x})\partial_{i}\phi(\bm{x})\,d^{3}x\right)$
(10)

where $\phi$ is given by Equation (4) and then we define the _product picture
physical subspace (PPPS)_ to be the result of acting with $U$ on the Coulomb
gauge physical subspace, i.e. as $U{\cal F}({\cal
H}_{\mathrm{one}}^{\mathrm{trans}})\otimes\Omega^{\mathrm{long}}\otimes{\cal
H}_{\mathrm{Dirac}}$. Let us notice here that all states in the PPPS
(including the vacuum state, $U\Omega$) are entangled between charged matter
and longitudinal photons!

We then have:

Theorem 1a. _$\tilde{\bm{\pi}}$ and $H^{\mathrm{PP}}_{\mathrm{QED}}$ (defined
as in (7) with the above definitions for $\hat{\bm{A}}$ and $\tilde{\pi}$) map
the PPPS to itself and are self-adjoint when restricted to it._

Theorem 1b. _Also the magnetic field operator,
$\bm{B}={\bm{\nabla}\times}\hat{\bm{A}}$, maps the PPPS to itself. (Although
the [as always, unphysical] vector potential, $\hat{\bm{A}}$ does not.)_

Theorem 2. _$H^{\mathrm{PP}}_{\mathrm{QED}}$ on the PPPS is (unitarily)
equivalent to $H_{\mathrm{QED}}^{\mathrm{Coulomb}}$ on the CGPS._

Theorem 3. _$\forall\,\bm{\Psi}\in$ PPPS,
$\bm{\nabla}\bm{\cdot}\bm{E}\,\bm{\Psi}\
(=-\bm{\nabla}\bm{\cdot}\tilde{\bm{\pi}}\,\bm{\Psi})=\rho\bm{\Psi}$, i.e.
Gauss’s law holds as an operator equation on the PPPS (in contrast to Coulomb
gauge, where it held by definition)._

We briefly sketch the proofs. For more details, see [8, Section 3].

_Proofs of Theorems 3 and 1a._ These follow easily after noting that
$U\tilde{\bm{\pi}}U^{-1}=\tilde{\bm{\pi}}-\bm{\nabla}\phi$

_Proof of Theorem 1b._ This follows immediately from the fact that
$\bm{\nabla\times}\hat{\bm{A}}$ is the same as $\bm{\nabla\times}$ acting on
the transverse part of $\hat{\bm{A}}$ (which is the same thing as the Coulomb
gauge $\bm{A}$) and that, as one easily sees, this maps the PPPS to itself.
(It is noteworthy, that the longitudinal part of $\hat{\bm{A}}$ _doesn’t_ map
the PPPS to itself but this is annihilated by $\bm{\nabla\times}$.)

_Proof of Theorem 2._ First define

$\check{H}_{\mathrm{QED}}=H_{\mathrm{QED}}^{\mathrm{Coulomb}}+\int\frac{1}{2}(\tilde{\bm{\pi}}^{\mathrm{long}})^{2}+\tilde{\bm{\pi}}\cdot\bm{\nabla}\phi\,d^{3}x$
(11)

and notice that, restricted to the CGPS,
$\check{H}_{\mathrm{QED}}=H_{\mathrm{QED}}^{\mathrm{Coulomb}}$.

Then on the full augmented QED Hilbert space, if one calculates
$U\check{H}_{\mathrm{QED}}U^{-1}$ one finds that it equals
$H_{\mathrm{QED}}^{\mathrm{PP}}$.

To do this calculation, it is helpful as a first step to notice that

$U\tilde{\bm{\pi}}U^{-1}=\tilde{\bm{\pi}}-\bm{\nabla}\phi,\quad U\psi
U^{-1}=\breve{\psi}$

where we define

$\breve{\psi}=e^{-\left(ie\frac{\partial_{i}}{\nabla^{2}}\hat{A}^{i}\right)}\psi.$

Using these, one easily finds that $U\check{H}_{\mathrm{QED}}U^{-1}$ is equal
to

$\int\frac{1}{2}\tilde{\bm{\pi}}^{2}+{1\over
2}({\bm{\nabla}}{\bm{\times}}{\bm{A}})^{2}+\breve{\psi}^{*}\gamma^{0}{\bm{\gamma}}{\bm{\cdot}}(-i{\bm{\nabla}}-{\rm
e}{\bm{A}})\breve{\psi}+m\breve{\psi}^{*}\gamma^{0}\breve{\psi}\,d^{3}x.$

It is then not difficult to see (or see [8, Section 3]) that this is equal to
the $H_{\mathrm{QED}}^{\mathrm{PP}}$ of (7). (End of proofs.)

Let us also remark that one may think of $\breve{\psi}$ as the appropriate
counterpart to $\psi$ in the product picture. In fact we have the dictionary
shown in the table.

Table 1: Coulomb gauge to product picture dictionary

Quantity | Coulomb gauge | product picture
---|---|---
Hamiltonian |  $\check{H}_{\mathrm{QED}}$ (or $H_{\mathrm{QED}}^{\mathrm{Coulomb}}$) | $H^{\mathrm{PP,Dirac}}_{\mathrm{QED}}$
electric field |  $\bm{E}_{\mathrm{C}}=-(\tilde{\bm{\pi}}+\bm{\nabla}\phi)$ (or $-(\bm{\pi}^{\perp}+\bm{\nabla}\phi)$) | ${\bm{E}}_{\mathrm{PP}}=-\tilde{\bm{\pi}}$
magnetic field |  $\bm{\nabla}\bm{\times}\bm{A}$ (or $\bm{\nabla}\bm{\times}{\bm{\hat{}}A}$) | $\bm{\nabla}\bm{\times}\hat{\bm{A}}$
Dirac field | $\psi$ | $\breve{\psi}=e^{-\left(ie\frac{\partial_{i}}{\nabla^{2}}\hat{A}^{i}\right)}\psi$
adjoint Dirac field | $\psi^{*}$ | $\breve{\psi}^{*}=e^{\left(ie\frac{\partial_{i}}{\nabla^{2}}\hat{A}^{i}\right)}\psi^{*}$
Dirac electrical potential | $\phi$ | $\phi$
vacuum state |  $\Omega^{\mathrm{trans}}\otimes\Omega^{\mathrm{long}}\otimes\Omega_{\mathrm{Dirac}}$ (or $\Omega^{\mathrm{trans}}\otimes\Omega_{\mathrm{Dirac}}$) |  $\Omega^{\mathrm{trans}}\otimes U(\Omega^{\mathrm{long}}\otimes\Omega_{\mathrm{Dirac}})$ (entangled)

Finally let us remark that there are two ways that we may obtain the unitary
time evolution $\exp(-iH_{\mathrm{QED}}^{\mathrm{PP}}t)$ on the PPPS (which
must both lead to the same answer). One way is to exponentiate ($-i$ times)
the self-adjoint restriction of $H_{\mathrm{QED}}^{\mathrm{PP}}t$ to the PPPS
on the PPPS. But another way would be to first exponentiate ($-i$ times) the
non-self-adjoint $H_{\mathrm{QED}}^{\mathrm{PP}}t$ on the augmented QED
Hilbert space – obtaining a nonunitary operator – and then to restrict that
nonunitary operator to the PPPS on which it will of course be unitary. In an
approach to constructing QED in a mathematically fully meaningful way based on
the product picture, the latter method may be technically advantageous, in
part because it would enable us to separate the issue of defining the dynamics
from the issue of assigning a mathematical meaning to the operator $U$ of
Equation (10) in Section 2 which defines the PPPS. This seems worthy of
further exploration.

##  3 The Contradictory Commutator Theorem and comparison with temporal gauge
quantization

The Hamiltonian, (7), resembles the classical Hamiltonian appropriate to the
_temporal gauge_ condition $A^{0}=0$. (Also known as the Weyl gauge
condition.) Several attempts have been made in the past to obtain a
formulation of QED by ‘quantizing’ that temporal gauge classical Hamiltonian –
taking the commutation relations to be those of Equation (8) but hitherto
certain assumptions have been (sometimes tacitly) made that led to pathologies
of one sort or another. (See, for example, [3].) For example one could define
$\hat{\bm{A}}$ in (8) as we have here but replace $\tilde{\bm{\pi}}$ by the
$\hat{\bm{\pi}}$ that we mentioned in Section 2. But one finds that any
attempt to obtain a theory equivalent to Coulomb gauge QED with those choices
would fail. The essence of what goes wrong can be summed up in what we shall
call the Contradictory Commutator Theorem. (See [8, Section 3.4] for more
details.)

The Contradictory Commutator (CC) Theorem. _There can be no pair of 3-vector
operators $\bm{\mathsf{A}}$ and $\bm{\mathsf{\pi}}$ on a Hilbert space
$\mathsf{H}$ such that_

(a) _$\bm{\mathsf{A}}$ and $\bm{\mathsf{\pi}}$ satisfy the canonical
commutation relations_

$[{\mathsf{A}}_{i}(\bm{x}),{\mathsf{\pi}}_{j}(\bm{y})]=i\delta_{ij}\delta^{(3)}(\bm{x}-\bm{y}),\
[{\mathsf{A}}_{i}(\bm{x}),{\mathsf{A}}_{j}(\bm{y})]=0=[{\mathsf{\pi}}_{i}(\bm{x}),{\mathsf{\pi}}_{j}(\bm{y})]$
(12)

(b) _$\bm{\mathsf{A}}$ and $\bm{\mathsf{\pi}}$ are each self-adjoint;_

(c) _For some vector $\bm{\Psi}\in\mathsf{H}$ ($\bm{\Psi}\neq 0$)_

$\bm{\nabla\cdot{\mathsf{\pi}}}\bm{\Psi}=-\rho\bm{\Psi}$

_for some operator-valued function of $\bm{x}$, $\rho$._

(d) _$\rho$ commutes with $\bm{\mathsf{A}}$._

_Proof._ (a) easily implies the equality

$\langle\bm{\Psi}|[{\mathsf{A}}_{i}(\bm{x}),\bm{\nabla\cdot{\mathsf{\pi}}}(\bm{y})]\bm{\Psi}\rangle=-i(\nabla_{i}\delta^{(3)})(\bm{x}-\bm{y}),$
(13)

while (b), (c) and (d) imply that quantity on the left hand side of (13) is
zero – a contradiction!

However our product picture evades the conclusions of this seeming no-go
theorem, as of course it must in view of Theorem 2. First let us notice that,
in attempting to apply the CC Theorem to our product picture, there are two
different interpretations we could make for the Hilbert space $\mathsf{H}$. We
could identify it with the full augmented QED Hilbert space, but then the CC
Theorem fails to apply because $\tilde{\bm{\pi}}$ fails to be self-adjoint on
$\mathsf{H}$. Alternatively, we could identify $\mathsf{H}$ with the product
picture physical subspace, and then $\tilde{\bm{\pi}}$ will map $\mathsf{H}$
to itself and be self adjoint on it. However $\hat{\bm{A}}$ will fail to map
$\mathsf{H}$ to itself! So either way the Contradictory Commutator Theorem
fails (as it must) to rule out our product picture formulation of QED.

###  3.1 The question of whether and to what extent the Contradictory
Commutator Theorem generalizes to Yang-Mills

It is natural to ask to what extent the formulation and proof of the
Contradictory Commutator Theorem generalize to Yang-Mills theory (say, for a
compact connected semisimple Lie group with Lie algebra generators, $T^{l}$,
satisfying $[T^{l},T^{m}]=if^{lmn}T^{n}$ for totally antisymmetric structure
constants $f^{lmn}$ and ${\rm tr}(T^{m}T^{n})=\tfrac{1}{2}\delta^{mn}$) and
also to gravity in general relativity.

We content ourselves here with a discussion of Yang-Mills theory (in flat
spacetime). Adopting signature $(+,-,-,-)$, we may take the classical
Lagrangian density of pure Yang-Mills to be $-\tfrac{1}{2}{\rm
tr}(F^{ab}F_{ab})$ where $F_{ab}$ is
$\nabla_{a}A_{b}-\nabla_{b}A_{a}-ig[A_{a},A_{b}]$ where $g$ is the Yang-Mills
coupling constant and $A_{a}=A_{a}^{l}T^{l}$. Choosing temporal gauge where
$A_{0}=0$, the conjugate momentum, say $\pi_{i}$, to $A_{i}$ is
$-F_{0i}=-\dot{A}_{i}$ while the counterpart to the Gauss law constraint is

$D_{i}\pi_{i}=-\rho$ (14)

where the covariant derivative, $D_{i}\pi_{i}$, written out in full, is given
by

$\nabla_{i}\pi_{i}-ig[A_{i},\pi_{i}].$ (15)

For, say, a quark-like field (assumed of course to commute with the Yang-Mills
fields) with values, $|\psi\rangle$, in the (finite dimensional) Hilbert space
of (say) the fundamental representation of the Lie algebra, $\rho$ is the
operator on this Hilbert space given by the formula (cf. (2))

$\rho=g|\psi^{*}\rangle\langle\psi|.$

The hypotheses of the CC theorem stated above for QED clearly have
counterparts for Yang-Mills fields which may be expressed with the same
equations provided one makes two changes in their interpretation. First we
need to reinterpret $\bm{\mathsf{A}}$ and $\bm{\mathsf{\pi}}$ as Lie-algebra
valued quantum (vector) fields where it is to be understood that (now
identifying our Lie algebra with its adjoint representation) next to the
‘$\delta_{ij}$’, there is an unwritten identity operator (on the Lie algebra
thought of as the representation vector space of the adjoint representation –
see below for a more accurate statement) on the right hand side of the
commutation relations in Part (a).

Secondly, the equation
$\bm{\nabla\cdot{\mathsf{\pi}}}\bm{\Psi}=-\rho\bm{\Psi}$ of Part (c) needs, in
light of (14), to be replaced by
$D_{i}\mathsf{\pi}_{i}\bm{\Psi}=-\rho\bm{\Psi}$ for a Lie-algebra valued
$\rho$ which commutes with $\bm{\mathsf{A}}$. (And the equation is still
assumed to hold for _some_ $\bm{\Psi}$ in the Hilbert space.)

With this changed interpretation of its hypotheses to make them relevant to
Yang-Mills fields, the question we want to ask, and will partly answer, is to
what extent the statement and proof of the theorem do – or don’t – go through
as before. We will also comment on the relation between our work and the work
in [1] and [3].

One big difference in the nonabelian case is that, because of the nonlinearity
of Yang-Mills theory, one now has to deal with products of fields at the same
point; notably in the second term in (15). However, adopting the point of view
of the first few pages of a paper [1] of Dimock, one may proceed by formally
manipulating expressions involving suitably smeared versions of our fields. To
begin to explain this, let us momentarily backtrack and notice that we might
have re-expressed the commutation relations (12) in the electromagnetic case
in terms of the smeared fields, $\bm{\mathsf{A}}(\bm{u})$ and
$\bm{\mathsf{\pi}}(\bm{v})$, where $\bm{u}$ and $\bm{v}$ are 3-vector valued
say smooth compactly supported test functions, which are to be interpreted in
terms of the formal equations

$\bm{\mathsf{A}}(\bm{u})=\int{\mathsf{A}}_{i}(\bm{x})u_{i}(\bm{x})\,d^{3}{\bm{x}},\quad\bm{\mathsf{\pi}}(\bm{v})=\int{\mathsf{\pi}}_{i}(\bm{x})v_{i}(\bm{x})\,d^{3}{\bm{x}}$

whereupon the first commutator in Equation (12) would have taken the form

$[\bm{\mathsf{A}}(\bm{u}),\bm{\nabla\cdot\mathsf{\pi}}(v)]=-i(\bm{u}|\bm{\nabla\cdot
v})$ (16)

where the symbol $(\,\cdot\,|\,\cdot\,)$ denotes the $L^{2}$ inner product in
the space of 3-vector fields.

Finally, let us remark that, since the unsmeared equations (13) needed a
‘$\delta_{ij}$’ on the right hand side due to the quantum vector fields being
expressed as triples of quantum scalar fields in some coordinate system, the
Hilbert space involved in that unsmeared version should have strictly been
taken to be the result of taking the tensor product of the Hilbert space
relevant to the smeared version with the (complexified) Hilbert space of
triples of numbers with the inner product the (complexified) dot product.

Returning to the Yang-Mills case, we proceed likewise but (following [1]) now
smearing $\mathsf{A}_{i}$ with a test triple $u_{i}$ whose components take
their values in the Lie algebra to obtain a smeared field $\bm{\mathsf{A}}(\bf
u)$ etc. where we think of the latter as “$({\mathsf{A}}_{i}|u_{i})$” (summed
over $i$ from 1 to 3) where the inner product, $(\,\cdot\,|\,\cdot\,)$ now
means the integral over $\mathbb{R}^{3}$ of minus the Killing form of
${\mathsf{A}_{i}}$ and $u_{i}$ (summed over $i$). (And similarly for
$\bm{\mathsf{\pi}}(\bf v)$.) And let us note, related to our above remark
about the ‘$\delta_{ij}$’, that when we reinterpreted the hypotheses of our CC
Theorem, written in their unsmeared version, to apply to Yang-Mills theories,
then, strictly, we should have thought of the Hilbert space as the tensor
product of the Hilbert space involved in the smeared version with the (again
finite dimensional) Hilbert space consisting of the complexified Lie Algebra
(thought of as a complex finite dimensional vector space) equipped with that
inner product. And the identity operator that we mentioned above should really
have been regarded as the identity operator on that latter Hilbert space.
(This is the more accurate statement about that identity operator that we
promised to make above.)

Assuming that the standard Lie-Algebra identity $([X,Y]|Z)=(X|[Y,Z])$ applies
also when some of $X$, $Y$, $Z$ are quantum operators, one easily sees
formally (again as in [1] but adapted to our conventions) that the smeared
quantity $(D_{i}{\mathsf{\pi}}_{i}|\bm{u})$ can be equated with
$(\bm{\mathsf{\pi}}|(\bm{\nabla\cdot
u})-g(\bm{\mathsf{A}}|[\bm{\mathsf{\pi}},\bm{u}])$ (or alternatively with
$(\bm{\mathsf{\pi}}|(\bm{\nabla\cdot
u})+g(\bm{\mathsf{\pi}}|[\bm{\mathsf{A}},\bm{u}])$). Making use of this, one
easily computes formally (now writing $\bm{\mathsf{A}}(\bm{u})$ in place of
$(\bm{\mathsf{A}}|\bm{u})$ etc.) that (cf. [1, Equation (8)])

$[\bm{\mathsf{A}}(\bm{u}),D_{i}\mathsf{\pi}_{i}(\bm{v})]=-i(\bm{u}|\bm{\nabla\cdot
v})-ig\bm{\mathsf{A}}([\bm{u},\bm{v}]).$ (17)

This is clearly the Yang-Mills counterpart to the commutator (16). But a
crucial difference is that, unlike the right hand side of (16), the right hand
side of (17) is no longer just a c-number but there is also an additional term
that involves the operator $\bm{\mathsf{A}}$. So, when we take the expectation
value of the above smeared commutator in a state vector, $\bm{\Psi}$, in the
appropriate Hilbert space, $\mathsf{H}$, the result will depend on $\bm{\Psi}$
and we will no longer obtain an immediate contradiction as we did in the proof
of the CC Theorem in the electromagnetic case. Instead, let us first shift our
viewpoint slightly and notice that one way of restating the CC Theorem in that
electromagnetic case is that, if (following [1]) we define a _physical state_
to be a state vector that satisfies $\bm{\nabla\cdot\pi\,\Psi}=0$, then the
set of physical states is empty! If we now attempt to adapt the steps of that
proof to the Yang-Mills case, by calculating the expectation value of the
commutator in the left hand side of (17) in the two different ways (one way
using Property (a), the other using Properties (b), (c) and (d)) we will
clearly find, now defining a physical state vector to be one satisfying
$D_{i}\pi_{i}\,\bm{\Psi}=0$ that if a state vector, $\bm{\Psi}$, is physical
then we must have that

$\langle\bm{\Psi}|\bm{\mathsf{A}}([\bm{u},\bm{v}])\bm{\Psi}\rangle=-g^{-1}(\bm{u}|\bm{\nabla\cdot
v}).$ (18)

Before we consider the implications of this, let us first notice that, in the
Yang-Mills case, there is also a non-trivial commutator between
$\bm{\mathsf{\pi}}(\bm{u})$ and $D_{i}\mathsf{\pi}_{i}(\bm{v})$ (cf. [1,
Equation (9)])

$[\bm{\mathsf{\pi}}(\bm{u}),D_{i}\mathsf{\pi}_{i}(\bm{v})]=i\bm{\mathsf{\pi}}([\bm{u},\bm{v}]).$
(19)

from which, by a similar argument, we may conclude that, for a state vector,
$\bm{\Psi}$, to be physical we must also have that, for all Lie-Algebra valued
smearing functions, $\bm{w}$,

$\langle\bm{\Psi}|\bm{\mathsf{\pi}}(\bm{w})\bm{\Psi}\rangle=0.$ (20)

(Here, we have used the fact that, in a semisimple Lie Algebra, every element
arises as the Lie bracket of two other elements.)

So we have the striking result that, for a state vector to be physical, the
expectation value of the field momentum $\bm{\mathsf{\pi}}$, which (in
components) is our quantized $-F_{0i}$ ($=-\dot{A}_{i})$, must vanish! On the
other hand, returning to the implications of (18), we clearly at least have
the immediate corollary that, for a state vector to be physical, the
expectation value of $\bm{\mathsf{A}}$ must _not_ vanish! More information
would seem to be extractable from (18) but we will not attempt to do so here.

The above results certainly seem to suggest that, in the case of Yang-Mills,
the set of physical state vectors (defined to be state vectors on which the
Yang-Mills counterpart to Gauss’s law in temporal gauge holds) is strangely
restricted. But this seems as far as one can easily go in directly
generalizing our CC theorem proof for nonabelian Yang-Mills. Note though that,
in the case of linearized (nonabelian) Yang-Mills, the right hand side of (17)
clearly becomes a c-number and so the proof goes through exactly as in the
electromagnetic case. Thus for linearized Yang-Mills we may conclude that the
set of physical states is empty!

Returning to full (nonlinear) Yang-Mills, in the later pages of Dimock’s
paper, defining $J(h)$ to be the Gauss law operator $D_{i}\pi_{i}$ smeared
with a test function $h$, he finds that $\bm{\mathsf{A}}$, $\bm{\mathsf{\pi}}$
and $J$ form an infinite dimensional Lie Algebra. Formally exponentiating that
to obtain an infinite dimensional Lie group, he points out that the elements
which formally correspond to $e^{iJ(h)}$ may be understood as the generators
of gauge transformations. The notion of ‘physical state’ is then redefined to
mean a state annihilated by all those elements. He then studies Hilbert space
representations of this Lie Group and finds classes of representations for
which it turns out that the set of physical state vectors, defined in that
way, is empty! However, using C* algebra techniques, he then finds a
representation which is a certain limit of one of those classes of
representations in which it is not empty. Presumably, in the electromagnetic
case, this coincides with the representation of the formal exponentiation of
Equations (12) that was found in [3] in which Gauss’s law holds strongly but
the fields $\bm{\mathsf{A}}$ and $\bm{\mathsf{\pi}}$ do not exist. (Rather
only their formal exponentials exist.)

On the other hand, for the electromagnetic case, we found here and in [8]
that, at least at our level of mathematical rigour, our product picture has
the advantage of evading the conclusions of the CC Theorem and in it the
fields $\bm{\mathsf{A}}$ and $\bm{\mathsf{\pi}}$ exist. Thus it would seem to
be of interest to find a suitable counterpart to our electromagnetic product
picture both for (linearized and full) Yang-Mills theory and also (for reasons
discussed further in Section 5) at least for linearized quantum gravity and,
as far as is possible, for full quantum gravity. And it would seem to be of
interest to attempt to make these product pictures mathematically rigorous.

##  4  A product picture for Maxwell-Schrödinger theory and the new
alternative understanding it entails for the nonrelativistic quantum mechanics
of many charged particles

There’s a similar product picture in the quantum electrodynamics of many non-
relativistic Schrödinger particles interacting with the EM field. (For more
details see [8, Section 4].) One has to treat the particles as finite radius
balls with charge density $\rho$. The Hamiltonian

$H^{\mathrm{Coulomb\,Schr}}_{\mathrm{QED}}=\int{1\over
2}{\bm{\pi}^{\perp}}^{2}+{1\over
2}({\bm{\nabla}}{\bm{\times}}{\bm{A}})^{2}d^{3}x+\sum_{I=1}^{N}\frac{\left(\bm{p}_{I}-\int{\bm{A}}(\bm{x})\rho_{I}(\bm{x}-\bm{x}_{I})\,d^{3}\bm{x}\right)^{2}}{2M_{I}}+V_{\mathrm{Coulomb}}^{\mathrm{Schr}}$
$\hbox{where}\quad
V_{\mathrm{Coulomb}}^{\mathrm{Schr}}=\frac{1}{2}\sum_{I=1}^{N}\sum_{J=1}^{N}\int\int\frac{\rho_{I}(\bm{x}-\bm{x}_{I})\rho_{J}(\bm{y}-\bm{x}_{J})}{4\pi|\bm{x}-\bm{y}|}\,d^{3}xd^{3}y$

on the appropriate CGPS gets replaced by

$H^{\mathrm{PP\,Schr}}_{\mathrm{QED}}=\int{1\over
2}\tilde{\bm{\pi}}^{2}+{1\over
2}({\bm{\nabla}}{\bm{\times}}\hat{\bm{A}})^{2}\,d^{3}x+\sum_{I=1}^{N}\frac{\left(\bm{p}_{I}-\int\hat{\bm{A}}(\bm{x})\rho_{I}(\bm{x}-\bm{x}_{I})\,d^{3}\bm{x}\right)^{2}}{2M_{I}}$
(21)

on the PPPS, all of whose states are again entangled between our charged balls
and longitudinal photons.

And we again have, defining $\check{H}^{\mathrm{Schr}}_{\mathrm{QED}}$ by
adding the same extra terms to $H^{\mathrm{Coulomb\,Schr}}_{\mathrm{QED}}$
that were added to $H^{\mathrm{Coulomb}}_{\mathrm{QED}}$ in (11),

$U\check{H}^{\mathrm{Schr}}_{\mathrm{QED}}U^{-1}=H^{\mathrm{PP\,Schr}}_{\mathrm{QED}},$

and

$U\bm{p}_{I}U^{-1}=\bm{p}_{I}-\int\hat{\bm{A}}^{\mathrm{long}}(\bm{x})\rho_{I}(\bm{x}-\bm{x}_{I})\,d^{3}x.$

etc.

In the usual approximation where we neglect terms which lead to radiative
corrections (and suppress the ${\cal F}({\cal
H}_{\mathrm{one}}^{\mathrm{trans}})\otimes$ in the Hilbert space) the Coulomb
gauge formulation of this nonrelativistic version of QED gives us the familiar
non-relativistic Schrödinger equation for systems of many charged particles.
(With extended balls, it is natural to include the [finite] self energies of
the balls in $V_{\mathrm{Coulomb}}^{\mathrm{Schr}}$ and in $E$.) So one might
wonder how the product picture can work in the nonrelativistic limit since the
Hamiltonian of Equation (21) has no potential term! To understand this, let us
look at the familiar example of the (spinless) Hydrogen atom. (For more
details, see [8, Section 4.2].) If Particle 1 is the proton and Particle 2 the
electron and the two-body wave function $\Psi_{\mathrm{Schr}}$ is say a
product of one of the usual Coulomb gauge energy eigenfunctions of the
relative coordinate with a wave packet in the centre of mass coordinate of
negligible energy, then in the usual Coulomb gauge description, we’ll have

$\left(\frac{\bm{p}_{1}^{2}}{2M_{1}}+\frac{\bm{p}_{2}^{2}}{2M_{2}}+V_{\mathrm{Coulomb}}^{\mathrm{Schr}}\right)\Omega^{\mathrm{long}}\otimes\Psi_{\mathrm{Schr}}\approx
E\Omega^{\mathrm{long}}\otimes\Psi_{\mathrm{Schr}}.$

In the product picture, this (on the CGPS) gets replaced by

$\left(\frac{\bm{p}_{1}^{2}}{2M_{1}}+\frac{\bm{p}_{2}^{2}}{2M_{2}}+{1\over
2}\mbox{$\tilde{\bm{\pi}}^{\mathrm{long}}$}^{2}\right)U\Omega^{\mathrm{long}}\otimes\Psi_{\mathrm{Schr}}\approx
EU\Omega^{\mathrm{long}}\otimes\Psi_{\mathrm{Schr}}.$

(on the PPPS). We see that the binding energy is now understood to be the
_decrease in the energy of the longitudinal part of the electric field when
the charged balls are closer to one another_. The (entangled) quantum state,
$U\Omega^{\mathrm{long}}\otimes\Psi_{\mathrm{Schr}}$, schematically takes the
form – when we think of ${\cal F}({\cal
H}_{\mathrm{one}}^{\mathrm{long}})\otimes L^{2}({\mathbb{R}}^{6})$ as
$L^{2}({\mathbb{R}}^{6},\,{\cal F}({\cal H}_{\mathrm{one}}^{\mathrm{long}}))$
–

$({\bm{x}}_{1},{\bm{x}}_{2})\mapsto\Psi_{\mathrm{Schr}}({\bm{x}}_{1},{\bm{x}}_{2})\Phi^{\mathrm{long}}({\bm{x}}_{1},{\bm{x}}_{2})$

where $\Phi^{\mathrm{long}}({\bm{x}}_{1},{\bm{x}}_{2})$ is the _coherent state
of longitudinal photons_ which corresponds to the classical electric field due
to the presence of the proton at ${\bm{x}}_{1}$ and the electron at
${\bm{x}}_{2}$.

For more information about exactly what is meant by coherent states of
longitudinal photons, see [8] which gives an alternative account of the
product picture formulation of QED which takes the construction of such
coherent states as its starting point. We also refer the reader to the paper
[8] for a discussion of a number of other topics including more about Gauss’s
law and the charged field commutator (see Section 3.3 there) and (in Section
4.3 there) about the reduced density operator of charged Schrödinger matter.
We shall also have more to say about the latter topic in the last section
below.

##  5  Afterword: The relevance to the matter gravity entanglement hypothesis
and comments on the relation with Penrose’s ideas about cosmological entropy
and state vector reduction

Aside from providing a new formulation of QED that is free from the
aesthetically unpleasant features that we mentioned at the outset, and aside
from offering the intriguing new way of thinking about the non-relativistic
quantum mechanics of many charged particles that we discussed in the Section
4, the product picture formulation of QED also permits us to ask some
questions about QED that could not even be posed in the Coulomb gauge
formulation. In particular, thanks to the fact that, in the product picture,
we can write the Hilbert space, ${\cal H}_{\mathrm{QED}}$, as a tensor product
of the form (5), it becomes meaningful to ask, for a (say pure) quantum state
of QED – thought of now as the density operator,
$\sigma_{\mathrm{QED}}=|\bm{\Psi}\rangle\langle\bm{\Psi}|$ for
$\bm{\Psi}\in{\cal H}_{\mathrm{QED}}$ – what is its reduced density operator
$\sigma_{\mathrm{charged\ matter}}$ on the Hilbert space ${\cal
H}_{\mathrm{charged\ matter}}$ (which we define to be the partial trace of
$\sigma$ over ${\cal H}_{\mathrm{electromag}}$). Similarly, we could ask what
is the reduced density operator $\sigma_{\mathrm{electromag}}$ on the Hilbert
space ${\cal H}_{\mathrm{electromag}}$ (which we define to be the partial
trace of $\sigma$ over ${\cal H}_{\mathrm{charged\ matter}}$). And, relatedly,
we can ask, e.g. what is the entanglement entropy of $\sigma_{\mathrm{QED}}$
between charged matter and the electromagnetic field. Here we recall that,
because of the assumed purity of $\sigma_{\mathrm{QED}}$, the von Neumann
entropy of $\sigma_{\mathrm{charged\ matter}}$ and the von Neumann entropy of
$\sigma_{\mathrm{electromag}}$ will be equal and their equal value is,
deservedly named the charged matter-electromag entanglement entropy of
$\sigma_{\mathrm{QED}}$.

Aside from any possible interest in these questions in their own right, they
are of interest as providing analogies to basic questions one asks on my
_matter gravity entanglement hypothesis_ (see [6, 7] and references therein)
which aims to help resolve a number of fundamental issues both about the
foundations of quantum mechanics (in particular regarding the possibility of
having an objective [observer-independent] notion of quantum state reduction)
and also about the foundations of thermodynamics (in particular regarding the
definition of entropy and the origin of the second law). This hypothesis was
inspired partly by the discovery [2] of the Hawking effect which suggests that
there may be new interconnections between quantum mechanics, gravity and
thermodynamics waiting to be discovered. It was also party inspired by Roger
Penrose’s writings (see for example [10]) on the status of the notion of
entropy in a cosmological context and the importance of understanding how the
entropy of the universe evolves in time from the big bang to the late-time
universe.

The idea of the matter-gravity entanglement hypothesis is that, to address
such questions, a closed system must be understood in the context of a
(possibly low-energy approximate) theory of quantum gravity which is assumed,
conservatively, to be formulated in terms of a total Hilbert space, say ${\cal
H}_{\mathrm{total}}$ which arises as a tensor product, ${\cal
H}_{\mathrm{gravity}}\otimes{\cal H}_{\mathrm{matter}}$ of a gravity Hilbert
space ${\cal H}_{\mathrm{gravity}}$ and a matter Hilbert space ${\cal
H}_{\mathrm{matter}}$. It is further assumed that the state of a closed system
is always a pure state, described by a density operator of form
$\sigma_{\mathrm{total}}=|\bm{\Psi}\rangle\langle\bm{\Psi}|$ for $\bm{\Psi}\in
H_{\mathrm{total}}$ and that (with a naïve notion of ‘time’ and from a
Schrödinger picture point of view) this evolves unitarily in time. A
fundamental process of decoherence (potentially offering an explanation of
quantum state reduction) will then be expected to occur if we assume that the
Hamiltonian generating our unitary time evolution (the quantum gravity
Hamiltonian) arises in the form (cf. Equation (7))

$H_{\mathrm{total}}=H_{\mathrm{gravity}}+H_{\mathrm{matter}}+H_{\mathrm{interaction}}$
(22)

and if we assume that the relevant density operator when we talk about the
occurence of decoherence is not $\sigma_{\mathrm{total}}$ itself (which will
always be pure) but rather the reduced density operator
$\sigma_{\mathrm{matter}}$. And we obtain a natural candidate for the physical
entropy of a closed system by defining it to be the system’s matter-gravity
entanglement entropy (rather than the von Neumann entropy of
$\sigma_{\mathrm{total}}$ which will always be zero). In other words the von
Neumann entropy of $\sigma_{\mathrm{matter}}$. (Or equally of
$\sigma_{\mathrm{gravity}}$.)

Indeed, suppose (now taking our closed system to be, say, the entire universe)
we make the further assumption that the initial state was unentangled between
matter and gravity (or perhaps just had a low degree of matter-gravity
entanglement) then, for any of a wide range of interaction terms,
$H_{\mathrm{interaction}}$, in (22), we would expect
$\sigma_{\mathrm{matter}}$ to get more and more mixed (thus providing us with
a fundamental process of decoherence) and concomitantly we would expect the
matter-gravity entanglement entropy (i.e. our model for the physical entropy
of the universe) to increase monotonically with time, thus offering the
possibility of an objective explanation for the second law.

Moreover, taking instead our closed system to be a model asymptotically flat
universe containing a ball of matter which collapses to a black hole, and
similarly assuming that the initial total state had a low degree of matter-
gravity entanglement, we would again expect $\sigma_{\mathrm{matter}}$ to get
more and more mixed and the system’s matter-gravity entanglement entropy to
increase monotonically. Thus we would reconcile ‘information loss’ (which we
interpret now as the reduced density operator $\sigma_{\mathrm{matter}}$
getting more and more mixed) with an underlying unitary evolution of
$\sigma_{\mathrm{total}}$.

Thus this very simple hypothesis already appears to give us a conceptual
template capable of both explaining the second law and resolving the black
hole information loss puzzle – which, in fact, is seen now just to be a
special case of the second law.

However, the entire hypothesis is predicated on quantum gravity having (to use
the terminology of our introduction) a product picture. And, even granting
that quantum gravity (or at least an appropriate low-energy approximation of
it which we might hope would be the suitable setting for discussing questions
of decoherence and entropy) will turn out to be a conservative sort of
(unitary) quantum theory, there might have seemed reasons to worry that this
quantum theory would not have a product picture. After all, one might have
objected, quantum gravity is, in a certain well-known sense, a gauge theory
and, more to the point, its Hamiltonian formulation involves four well-known
constraints. And one might then have argued by analogy that, just as the Gauss
law constraint prevents quantum electrodynamics, in its Coulomb gauge
formulation, from having a product picture, so one might expect that the four
constraints of general relativity will prevent quantum gravity from having a
product picture.

However, now that we have provided QED with an alternative formulation which
is a product picture, this worry is at least to some extent dispelled. It
raises the hope that, with further work, we will similarly be able to provide
quantum gravity with a product picture.

To end this afterword, I would like to make some tentative and sketchy remarks
about how our matter gravity entanglement hypothesis might possibly be related
to Roger Penrose’s ideas (e.g. as discussed in [10]) about cosmological
entropy and especially to his related ideas (see e.g. [11]) about quantum
state reduction. It seems to me to be still quite unclear at present whether
the two lines of thought are fully in contradiction with one another or
whether our hypothesis perhaps stands in relation to Penrose’s vision in
somewhat the same way that, e.g., Bohr’s 1910 theory of the Hydrogen atom
related to the 1924 ideas of de Broglie (which led, a little later, to the
Heisenberg-Schrödinger theory of quantum mechanics).

One point of contact is that both theories have addressed the issue of what
happens to would-be macroscopic quantum superpositions involving different
spacetime geometries and, for concreteness, both theories have been
illustrated by considering a Schrödinger Cat-like superposition of two states
of a single uniform density massive ball centred on two distinct locations in
an otherwise empty (asymptotically flat) space. Each ball state will have its
own (Newtonian) gravitational field and the question is whether and in what
sense the total state might spontaneously decohere. According to Penrose’s
ideas [11] one expects such a macroscopic superposition to decay to a state
centred on a single location on some time-scale which depends on the size and
mass of the ball and the separation of the centres of mass of the two ball
states in the superposition. According to the matter gravity entanglement
hypothesis, such a superposition could be in a static state but the relevant
density operator, when one asks questions about purity versus mixedness, will
not be its $\sigma_{\mathrm{total}}$ (which will always be pure) but rather it
will be the reduced density operator $\sigma_{\mathrm{matter}}$. One can
compute this similarly to the way in which one would compute
$\sigma_{\mathrm{charged\ matter}}$ in our product picture for QED for a
superposition involving two states of an electrically charged ball centred on
two different locations in the non-relativistic version of QED described in
Section 4. In fact formulae for both $\sigma_{\mathrm{charged\ matter}}$ in
the QED analogy and for $\sigma_{\mathrm{matter}}$ in weak-field Newtonian
quantum gravity were obtained by the author in [4] (albeit I was able, only
around 24 years later, to provide them, in the case of QED, with the more
satisfactory theoretical underpinning described here and in [8] [where the
formulae are reproduced in Sections 1 and 4.3] and I should mention that work
is in progress on providing such an underpinning [and the fixing of an overall
constant] in the case of linearized gravity) and one finds that (in the case
the displacement of the centres is much smaller than the ball radius) the
degree of mixedness of $\sigma_{\mathrm{charged\
matter}}$/$\sigma_{\mathrm{matter}}$ is governed by a certain _decoherence
length_ which again depends on the size and mass of the ball and the
separation of the centres of mass of the two ball states in the superposition.

As discussed in more detail in [4] the formula obtained there for the
decoherence length is remarkably similar to the formula obtained (up to a
presently unfixable multiplicative constant) by Penrose in [11] for his decay
time-scale. (The only difference is that where there is a Laplacian in the
latter formula, there is a square root of the Laplacian in the former.) On the
other hand there are important physical differences. One difference, as we
have already seen, is that the decoherence is a time-dependent process in the
Penrose theory while it is ever present in the setting of [4]. Another is that
the sort of decoherence one obtains on the theory of [8, 4] is, in the case of
static or slowly changing ball configurations, a reversible sort of
decoherence; if one were to (slowly) bring the ball centres back into
coincidence, the decohered ball states would recohere. For this reason, in the
Newtonian limit, the matter gravity entanglement hypothesis appears (see [5]
for details) to predict the same result as standard quantum mechanics for the
experiment [9] proposed by Penrose and thus to be experimentally
distinguishable from his theory. One expects instead that, on the matter-
gravity entanglement hypothesis, irreversible decoherence would occur and lead
to a noticeable loss of coherence in that experiment only if the oscillator in
the experiment were to oscillate sufficiently rapidly to emit a graviton in a
suitably short time interval – a much harder regime to access. (For more
discussion and for more discussion of the physical interpretation of
$\sigma_{\mathrm{matter}}$ see again [5] and [7].)

Another point of difference between the two sets of ideas is that, on the
matter-gravity entanglement hypothesis, the entropy of the universe resides at
all times in the entanglement of matter with gravity (which is assumed to
start low and then predicted to increase) whereas, on Penrose’s ideas, one
apparently thinks in terms of an entropy which is a sum of a matter entropy
(which dominates at early times) and a gravity entropy (which dominates at
late times and is larger than the matter entropy at early times). Here there
appears to be an antagonism between the two theories which resembles (and
partly subsumes) the antagonism between the understanding of black hole
entropy on the matter-gravity entanglement hypothesis (according to which the
total state is pure while it is the reduced density operator of matter as well
as the reduced density operator of gravity which are approximately Gibbs
states) and the traditional (Hawking) understanding of black holes according
to which the total state is a (mixed) Gibbs state. See the proposed resolution
of the _thermal atmosphere puzzle_ in [6].

## Acknowledgments

The author thanks the Leverhulme Foundation for the award of Leverhulme
Fellowship RF&G/9/RFG/2002/0377 for the period October 2002 to June 2003
during which some of this work was done. I thank Jonathan Dimock and Michael
Kay for valuable comments.

## Data Availability

Data sharing is not applicable to this article as no new data were created or
analyzed in this study.

## Author Declaration

The author has no conflicts to disclose.

## References

  * [1] Dimock, J.: Canonical quantization of Yang-Mills on a circle. Reviews in Mathematical Physics. 8 85-102 (1996)
  * [2] Hawking, S.W.: Particle creation by black holes. Communications in Mathematical Physics, 43 199-220 (1975)
  * [3] Löffelholz, J., Morchio, G., Strocchi, F.: Mathematical structure of the temporal gauge in quantum electrodynamics. J. Math. Phys. 44, 5095-5107 (2003) [arXiv:math-ph/0212039]
  * [4] Kay, B.S.: Decoherence of macroscopic closed systems within Newtonian quantum gravity. Class. Quant. Grav. 15 L89-L98, (1998) [arXiv:hep-th/9810077]
  * [5] Kay, B.S.: Expectation values, experimental predictions, events and entropy in quantum gravitationally decohered quantum mechanics. arXiv:0710.0992 (2007)
  * [6] Kay, B.S.: The matter gravity entanglement hypothesis. Foundations of Physics. 48 542–557 (2018) [arXiv:1802.03635]
  * [7] Kay, B.S.: Remarks on matter-gravity entanglement, entropy, information loss and events. In: Finster F., Giulini D., Kleiner J., Tolksdorf J. (eds) Progress and Visions in Quantum Theory in View of Gravity. Birkhäuser, Cham (2020)
  * [8] Kay, B.S.: Quantum electrostatics, Gauss’s law, and a product picture for quantum electrodynamics; or, the temporal gauge revised. Foundations of Physics. 52: 6 (2022) [arXiv:2003.07473]
  * [9] Marshall, W., Simon, C., Penrose, R., Bouwmeister, D.: Towards quantum superpositions of a mirror. Phys. Rev. Lett. 91 130401 (2003); Erratum Phys. Rev. Lett. 91 159903 (2003)
  * [10] Penrose, R.: Singularities and time asymmetry. In: Hawking, S.W., Israel, W. (eds): General Relativity: An Einstein Centenary Survey, Cambridge University Press, Cambridge (1979)
  * [11] Penrose. R.: On gravity’s role in quantum state reduction. General relativity and gravitation, 28, 581-600 (1996)
  * [12] Weinberg, S.: The Quantum Theory of Fields, Volume 1. Cambridge University Press, Cambridge (1995)


[1]

<EMAIL_ADDRESS>
# Stellar $\beta$-decay rate of $s$-process branching-point 204Tl: forbidden
transitions

Yang Xiao  School of Physical Science and Technology, Southwest University,
Chongqing 400715, China  Bin-Lei Wang  Long-Jun Wang

###### Abstract

We propose a theoretical method to calculate the stellar $\beta$-decay rates
of nuclei in stellar environments with high temperature and density, based on
the projected shell model, where contributions from both allowed and first-
forbidden transitions are taken into account. As the first example, the
stellar $\beta$-decay rate of one of the last $s$-process branching-point
nuclei, 204Tl, is calculated and studied, where all related transitions are
first-forbidden transitions. For the terrestrial case, the ground-state to
ground-state transition is unique first-forbidden transition, which is
described reasonably by our calculations. At the typical $s$-process
temperature ($T\approx 0.3$ GK), non-unique first-forbidden transitions from
thermally populated excited states of the parent nucleus are involved, the
effective rate from our calculations is much lower than the one from the
widely used data tables by Takahashi and Yokoi. Effect of the quenching
factors for nuclear matrix elements in first-forbidden transitions on the
stellar $\beta$-decay rates is discussed as well.

###### keywords:

Stellar $\beta$ decay $s$ process Branching-point nuclei First-forbidden
transition Projected shell model

Nuclear $\beta$ decay plays crucial and indispensable roles in many aspects in
nuclear astrophysics [1, 2, 3, 4, 5, 6]. The stellar $\beta$-decay rates in
stellar environments with high temperature and high density, are important
inputs for understanding many astrophysical problems regarding the evolution
of stars and the origin of heavy elements, such as the core-collapse supernova
of massive stars, the slow ($s$-) and rapid ($r$-) neutron-capture processes
[7, 8], the rapid proton-capture ($rp$-) process [9], the cooling of neutron
stars [10] etc. In these problems, a lot of exotic nuclei with very short
half-lives and/or transitions from excited states due to the thermal
population are included, which are challenging for current experimental
measurements. Therefore, theoretical calculations from different nuclear many-
body methods are relied on heavily, and reliable nuclear inputs become
increasingly important.

The $s$ process contributes approximately half of the solar elemental
abundances between Fe and Bi [7]. The $s$ process is expected to take place
during core He-burning and shell C-burning of massive stars (weak $s$ process)
as well as in H-burning and He-burning layers of the Asymptotic Giant Branch
(AGB) phase of low-mass stars (main $s$ process), with the peak temperature
(density) reaching $\approx 0.3$ GK ($\approx 10^{10}$ cm-3). An important and
interesting problem during the $s$ process is the branching point, which
refers to unstable isotope with its neutron-capture reaction rate can compete
with corresponding $\beta$-decay rate [7, 11]. The branching-point nuclei play
decisive roles in determining the reaction flow, the path, the final
abundances etc. due to the competition between the two branches, further
neutron capture and $\beta^{-}$ decay, where the effective $\beta$-decay rates
in $s$-process environments are indispensable inputs.

Figure 1:  (color online) An excerpt of the chart of nuclides near the
termination of the $s$ process, where the branching-point 204Tl is
illustrated. Neutron capture, $\beta$ decay and electron capture (EC) are
labeled by arrows.

An example of the $s$-process branching points, 204Tl, is shown in Fig. 1. On
one hand, 204Tl is one of the two last branching points near the termination
of the $s$ process (the other is 205Tl for which bound-state $\beta^{-}$ decay
is open when temperature $T\gtrsim 0.16$ GK) [12, 13, 14, 15, 16, 17, 18, 19].
The relevant nuclei, 204Pb and 205Pb, are $s$-only nuclei which can be
produced only by the $s$ process, so that they are potential chronometers of
the last $s$-process events and they become probes of the corresponding
stellar conditions. It is seen from Fig. 1 that the $\beta^{-}$ decay of 204Tl
is crucial to determine the final abundance of 204Pb. On the other hand, the
stellar $\beta^{-}$-decay rate of 204Tl is predicted to depend sensitively on
temperature, leading to 204Tl as an $s$-process thermometer. Besides, nuclear
allowed transitions usually dominate the stellar $\beta^{-}$ decays of
branching points, while in many cases such as for 151Sm, 170Tm, 185W, 204Tl
etc. the stellar $\beta^{-}$ decay rates are determined by first-forbidden
transitions, where 204Tl serves as a special example as all the related
transitions are first-forbidden.

Generally the stellar decay rates from the data tables by Takahashi and Yokoi
(referred to as TY87 hereafter) [12] are adopted for studies of $s$ process
(branching points). While the TY87 data are outdated as transitions from
excited states are neither from experimental measurements nor from
sophisticated nuclear many-body calculations, but determined with the
corresponding comparative half-life, $ft$, estimating empirically by similar
transitions of neighboring nuclei with available data (usually from ground
states). The TY87 data regarding transitions from excited states have been
challenged by modern charge-exchange reactions (for 59Fe [20]) or shell-model
calculations (for 134Cs [21]) recently. Over the past decade, the projected
shell model (PSM) [22, 23] is extended to have large configuration space
considering high-order quasiparticle (qp) configurations with the help of the
Pfaffian and other algorithms [24, 25, 26, 27, 28]. The PSM is then applied to
studies on nuclear high-spin physics [26, 29, 30]. Besides, the PSM is further
developed for describing allowed Gamow-Teller (GT) transition and stellar
weak-interaction rates [31, 32, 33, 34, 35, 36, 37] and nuclear $\beta$
spectrum [38] recently. A new development of the PSM for description of first-
forbidden transition of nuclear $\beta$ decay is accomplished very recently
[39]. It is then straightforward to present a PSM method for stellar weak-
interaction rates with both allowed GT and first-forbidden transitions taken
into account here, and taking the branching-point 204Tl as an example.

Within the common assumption that parent nuclei are in a thermal equilibrium
with occupation probability for excited states following the Boltzmann
distribution, the stellar $\beta^{-}$ decay rates read as [1, 2, 3, 4],

$\displaystyle\lambda^{\beta^{-}}=\sum_{if}\frac{(2J_{i}+1)e^{-E_{i}/(k_{B}T)}}{G(Z,A,T)}\lambda^{\beta^{-}}_{if},$
(1)

where the summations run over initial ($i$) and final ($f$) states of parent
and daughter nuclei, respectively (with spin-parity assignments
$J_{i}^{\pi_{i}},J_{f}^{\pi_{f}}$ and excitation energies $E_{i},E_{f}$).
$k_{B}$ labels the Boltzmann constant and $T$ is the environment temperature.
$G(Z,A,T)=\sum_{i}(2J_{i}+1)\text{exp}(-E_{i}/(k_{B}T))$ is the partition
function. With $G(Z,A,T)$ as the denominator, the factor reflects the thermal
population probability of the initial (low-lying) state. The individual rate,

$\displaystyle\lambda^{\beta^{-}}_{if}=$ $\displaystyle\frac{\ln
2}{K}\int_{1}^{Q_{if}}C(W)F_{0}(Z+1,W)pW(Q_{if}-W)^{2}$
$\displaystyle\qquad\quad\times(1-S_{e}(W))dW,$ (2)

where the constant $K$ can be determined from superallowed Fermi transitions
and $K=6144\pm 2\ s$ [40] is adopted in this work. $W$ and $p=\sqrt{W^{2}-1}$
label the total energy (rest mass and kinetic energy) and the momentum of the
electron in units of $m_{e}c^{2}$ and $m_{e}c$ respectively. The available
total energy for leptons in the one-to-one $\beta^{-}$ transition is given by,

$\displaystyle Q_{if}=\frac{1}{m_{e}c^{2}}(M_{p}-M_{d}+E_{i}-E_{f}),$ (3)

where $M_{p}(M_{d})$ indicates the nuclear mass of parent (daughter) nucleus.
In Eq. (Stellar $\beta$-decay rate of $s$-process branching-point 204Tl:
forbidden transitions), $S_{e}$ is the electron distribution function
following the Fermi-Dirac distribution as,

$\displaystyle S_{e}(W)=\frac{1}{\text{exp}[(W-\mu_{e})/k_{B}T]+1},$ (4)

in which the chemical potential, $\mu_{e}$, is determined from the relation,

$\displaystyle\rho
Y_{e}=\frac{1}{\pi^{2}N_{A}}\left(\frac{m_{e}c}{\hbar}\right)^{3}\int_{0}^{\infty}(S_{e}-S_{p})p^{2}dp,$
(5)

here $N_{A}$ represents Avogadro’s number, and $\rho Y_{e}$ labels the
electron density. Note that one can get the positron distribution $S_{p}$ by
the replacement $\mu_{p}=-\mu_{e}$. In Eq. (Stellar $\beta$-decay rate of
$s$-process branching-point 204Tl: forbidden transitions) $F_{0}(Z+1,W)$ is
the Fermi function that reveals the Coulomb distortion of the electron wave
function near the nucleus [1, 41], with $Z$ being the proton number of the
decaying (parent) nucleus.

The $C(W)$ term in Eq. (Stellar $\beta$-decay rate of $s$-process branching-
point 204Tl: forbidden transitions) is the shape factor for the decay. In the
case of allowed transitions with selection rule $|\Delta
J|=|J_{i}-J_{f}|=0,1$, and $\Delta\pi=\pi_{i}\pi_{f}=+1$, $C(W)$ does not
depend on the electron energy [42], and can be written as,

$\displaystyle
C(W)=B(\text{GT}^{-})_{if}=\left(\frac{g_{A}}{g_{V}}\right)^{2}_{\text{eff}}\frac{\big{\langle}\Psi_{J_{f}}^{n_{f}}\big{\|}\sum_{k}\hat{\bm{\sigma}}^{k}\hat{\tau}_{-}^{k}\big{\|}\Psi_{J_{i}}^{n_{i}}\big{\rangle}^{2}}{2J_{i}+1},$
(6)

where only GT contribution is considered as usually [43, 44, 45]. Here
$\hat{\bm{\sigma}}$ ($\hat{\tau}_{-}$) is the Pauli spin operator (isospin
lowering operator), the summation runs over all nucleons, and $\Psi_{J}^{n}$
represents the $n$-th eigen nuclear many-body wave function for angular
momentum $J$. $(g_{A}/g_{V})_{\text{eff}}$ is the effective ratio of axial and
vector coupling constants with corresponding quenching for the GT matrix
element (operator and/or wave function) [46, 47, 48, 49],

$\displaystyle\left(\frac{g_{A}}{g_{V}}\right)_{\text{eff}}=f_{q}(\text{GT})\left(\frac{g_{A}}{g_{V}}\right)_{\text{bare}},$
(7)

where $(g_{A}/g_{V})_{\text{bare}}=-1.27641(45)$ [50], and $f_{q}(\text{GT})$
is the quenching factor for GT transition.

For first-forbidden transitions, $C(W)$ has explicit energy dependence, which
is approximated as [51, 42, 52]

$\displaystyle C(W)=k(1+aW+b/W+cW^{2}),$ (8)

for non-unique transitions with $|\Delta J|=0,1$, $\Delta\pi=-1$, and [52]

$\displaystyle C(W)=k(1+aW+b/W+cW^{2})\left[q^{2}+\lambda_{2}p^{2}\right],$
(9)

for unique transitions with $|\Delta J|=2$, $\Delta\pi=-1$, where $q=Q_{if}-W$
is the energy of the neutrino and $\lambda_{2}=\frac{F_{1}(Z,W)}{F_{0}(Z,W)}$
with $F_{1}$ being the generalized Fermi function [53]. The coefficients
$k,a,b$ and $c$ depend on the following reduced nuclear matrix elements for
first-forbidden transition (see Refs. [42, 39] for the detailed dependence),

$\displaystyle w$ $\displaystyle=$ $\displaystyle-
g_{A}\sqrt{3}\frac{\left\langle\Psi^{n_{f}}_{J_{f}}\left\|\sum_{k}r_{k}[\bm{C}^{k}_{1}\otimes\hat{\bm{\sigma}}^{k}]^{0}\hat{\tau}^{k}_{-}\right\|\Psi^{n_{i}}_{J_{i}}\right\rangle}{\sqrt{2J_{i}+1}},$
(10a) $\displaystyle x$ $\displaystyle=$
$\displaystyle-\frac{\left\langle\Psi^{n_{f}}_{J_{f}}\left\|\sum_{k}r_{k}\bm{C}^{k}_{1}\hat{\tau}^{k}_{-}\right\|\Psi^{n_{i}}_{J_{i}}\right\rangle}{\sqrt{2J_{i}+1}},$
(10b) $\displaystyle u$ $\displaystyle=$ $\displaystyle-
g_{A}\sqrt{2}\frac{\left\langle\Psi^{n_{f}}_{J_{f}}\left\|\sum_{k}r_{k}[\bm{C}^{k}_{1}\otimes\hat{\bm{\sigma}}^{k}]^{1}\hat{\tau}^{k}_{-}\right\|\Psi^{n_{i}}_{J_{i}}\right\rangle}{\sqrt{2J_{i}+1}},$
(10c) $\displaystyle z$ $\displaystyle=$ $\displaystyle
2g_{A}\frac{\left\langle\Psi^{n_{f}}_{J_{f}}\left\|\sum_{k}r_{k}[\bm{C}^{k}_{1}\otimes\hat{\bm{\sigma}}^{k}]^{2}\hat{\tau}^{k}_{-}\right\|\Psi^{n_{i}}_{J_{i}}\right\rangle}{\sqrt{2J_{i}+1}},$
(10d) $\displaystyle w^{\prime}$ $\displaystyle=$ $\displaystyle-
g_{A}\sqrt{3}\frac{\left\langle\Psi^{n_{f}}_{J_{f}}\left\|\sum_{k}\frac{2}{3}r_{k}I(r_{k})[\bm{C}^{k}_{1}\otimes\hat{\bm{\sigma}}^{k}]^{0}\hat{\tau}^{k}_{-}\right\|\Psi^{n_{i}}_{J_{i}}\right\rangle}{\sqrt{2J_{i}+1}},$
$\displaystyle x^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{\left\langle\Psi^{n_{f}}_{J_{f}}\left\|\sum_{k}\frac{2}{3}r_{k}I(r_{k})\bm{C}^{k}_{1}\hat{\tau}^{k}_{-}\right\|\Psi^{n_{i}}_{J_{i}}\right\rangle}{\sqrt{2J_{i}+1}},$
(10f) $\displaystyle u^{\prime}$ $\displaystyle=$ $\displaystyle-
g_{A}\sqrt{2}\frac{\left\langle\Psi^{n_{f}}_{J_{f}}\left\|\sum_{k}\frac{2}{3}r_{k}I(r_{k})[\bm{C}^{k}_{1}\otimes\hat{\bm{\sigma}}^{k}]^{1}\hat{\tau}^{k}_{-}\right\|\Psi^{n_{i}}_{J_{i}}\right\rangle}{\sqrt{2J_{i}+1}},$
$\displaystyle\xi^{\prime}\nu$ $\displaystyle=$
$\displaystyle\frac{g_{A}\sqrt{3}}{M_{0}}\frac{\left\langle\Psi^{n_{f}}_{J_{f}}\left\|\sum_{k}[\hat{\bm{\sigma}}_{k}\otimes\bm{\nabla}^{k}]^{0}\hat{\tau}^{k}_{-}\right\|\Psi^{n_{i}}_{J_{i}}\right\rangle}{\sqrt{2J_{i}+1}},$
(10h) $\displaystyle\xi^{\prime}y$ $\displaystyle=$
$\displaystyle-\frac{1}{M_{0}}\frac{\left\langle\Psi^{n_{f}}_{J_{f}}\big{\|}\sum_{k}\bm{\nabla}^{k}\hat{\tau}^{k}_{-}\big{\|}\Psi^{n_{i}}_{J_{i}}\right\rangle}{\sqrt{2J_{i}+1}}.$
(10i)

where the definitions of $\bm{C}_{lm}$, the radial function $I(r)$ etc. can be
seen from Refs. [42, 39].

As in the allowed GT transition of $\beta$ decay [46, 47] and in the double
$\beta$ decay [48, 49], quenching factors for nuclear matrix elements
(operators and/or wave functions) of first-forbidden transitions in Eq. (10)
should probably be introduced as well [42]. In the shell-model studies of Ref.
[42] the following quenching factors are adopted,

$\displaystyle f_{q}(\xi^{\prime}\nu)$ $\displaystyle=1.266,\hskip
39.50002ptf_{q}(w)=f_{q}(w^{\prime})=0.66,$ $\displaystyle f_{q}(x)$
$\displaystyle=f_{q}(x^{\prime})=0.51,\quad f_{q}(u)=f_{q}(u^{\prime})=0.38,$
$\displaystyle f_{q}(z)$ $\displaystyle=0.42.$ (11)

Figure 2:  (color online) The calculated excitation energies for low-lying
states of the parent and daughter nuclei, as compared with available data
[54].

For both the allowed transitions as in Eq. (6) and the first-forbidden
transitions as in Eq. (10), nuclear many-body wave functions in the laboratory
frame, $\Psi_{J}^{n}$, from elaborate nuclear models are necessary for
calculations. In this work, $\Psi_{J}^{n}$ is described by the PSM [25, 26],
and the corresponding allowed GT transitions and first-forbidden transitions
are fulfilled based on the recent developments [32, 33, 38, 39], from which
the calculation of stellar $\beta$-decay rates with both allowed GT and first-
forbidden transitions taken into account can be accomplished. The $s$-process
branching-point 204Tl is taken as the example for discussions.

Figure 2 illustrates the calculated energy levels for the parent 204Tl and
daughter 204Pb nuclei as compared with the data [54]. It is seen that the data
are described reasonably. In particular, the spin and parity of the ground
state for the odd-odd nucleus 204Tl is well reproduced. Besides, the energy
gap between the ground state and the excited states for 204Pb is described
reasonably, and the behavior of dense levels for low-lying states of 204Tl is
presented. The first and second excited states of 204Tl are preset tentatively
to have spin-parity $1^{-}$ and $0^{-}$, which is supported by our PSM
calculations. The experimental excitation energies for these two low-lying
states are 140 keV and 146 keV respectively, which are difficult to be
thermally populated effectively at the typical $s$-process temperature
$T\approx 0.3$ GK. However, provided that the $\beta$-decay $Q$ value of 204Tl
is as small as 764 keV, and because that the transitions from these two low-
lying states are predicted in the TY87 data tables [12] to be much stronger
than the terrestrial transition between ground states by about five orders of
magnitude, as seen from Fig. 3, even tiny thermal population probability of
them could increase rapidly the stellar $\beta$ decay rate of 204Tl with the
temperature, so that leading to 204Tl as an $s$-process thermometer and
affecting the potential chronometers of the last $s$-process events, 205Pb.

Figure 3:  (color online) The schematic stellar $\beta$-decay scheme of 204Tl.
Calculated Log$ft$ values without (in green brace) and with (in red
parenthesis) the quenching factors in Eq. (Stellar $\beta$-decay rate of
$s$-process branching-point 204Tl: forbidden transitions) are compared with
the TY87 results (in blue bracket) [12] for transitions from low-lying states,
and the data [54] for transition between ground states (gs). Excitation
energies are in keV.

Due to the importance as discussed above, it is necessary to study the stellar
$\beta$-decay rate of 204Tl within elaborate nuclear many-body method, since
the unknown transition strengths from excited states are estimated empirically
in the TY87 data tables [12]. Figure 3 shows the schematic $\beta$-decay
scheme of 204Tl in the $s$-process environments. In terrestrial case, the
204Tl stays in the ground state and its half-life is 3.783 yr, which is
dominated by the $\beta^{-}$-decay channel with the EC channel contributing
less than 3$\%$, so that we focus on the $\beta^{-}$-decay channel in the
following. Owing to the small $Q$ value, only ground-state to ground-state
transition is relevant for the $\beta^{-}$ decay of 204Tl to 204Pb in
terrestrial case. Such a transition is unique first-forbidden and is found
experimentally to be much weak with Log$ft=10.10$. It is seen from Fig. 3 that
our PSM calculations present stronger transition strength than the data with
Log$ft=9.59$ when no quenching factor is adopted. The data can be described
reasonably by our PSM calculations when the quenching factors in Eq. (Stellar
$\beta$-decay rate of $s$-process branching-point 204Tl: forbidden
transitions) are adopted with Log$ft=10.34$.

With the increase of temperature during the $s$ process, the first and second
excited states of 204Tl may be populated thermally. The transitions from these
two low-lying states then become relevant, which are non-unique first-
forbidden and the transition strengths are expected to be much strong with
Log$ft\approx 5$ by the TY87 data tables [12] as seen from Fig. 3. By
comparison, the corresponding transition strengths predicted by the PSM
calculations are reduced largely and turn out to be dependent sensitively on
the quenching. On one hand, this indicates that the stellar $\beta^{-}$-decay
rate of 204Tl in the $s$-process environments may be reduced largely than the
one from the widely used TY87 data tables. On the other hand, the quenching
effect of nuclear matrix elements for first-forbidden transitions is important
in our understanding of the roles played by first-forbidden transitions in the
stellar weak-interaction rates.

Figure 4:  (color online) (a) The calculated stellar $\beta^{-}$-decay rate of
204Tl by the PSM with and without the quenching factors in Eq. (Stellar
$\beta$-decay rate of $s$-process branching-point 204Tl: forbidden
transitions) as a function of the stellar temperature, compared with the TY87
rate [12] and the constant terrestrial rate. (b) The partial decay rates of
individual transitions by the PSM calculations with the quenching factors,
compared to the total one. See the text for details.

With the experimental energy levels and the calculated shape factors $C(W)$
for the three first-forbidden transitions in Fig. 3 by the PSM, the calculated
stellar $\beta^{-}$-decay rate of 204Tl is compared with the TY87 rate and the
terrestrial one in Fig. 4(a), where the electron number density is adopted as
$\rho Y_{e}=1.66\times 10^{2}$ mol/cm3 in the PSM calculations (this
corresponds to $n_{e}=10^{26}$ cm-3 in Ref. [12] and the rate of 204Tl shows
almost no sensitive dependence on electron density in our calculations and in
Ref. [12]). It is seen that as the $0^{-}\rightarrow 0^{+}$ transition is
evaluated empirically to be stronger than the ground-state to ground-state one
by five orders of magnitude (Fig. 3), the TY87 rate increases very rapidly
when the temperature $T\geqslant 0.1$ GK. Such a sensitive temperature
dependence make 204Tl as a potential $s$-process thermometer. By comparison,
the calculated rate by the PSM without (with) the quenching factors in Eq.
(Stellar $\beta$-decay rate of $s$-process branching-point 204Tl: forbidden
transitions) is higher (a little reduced) than the terrestrial data at
$T\lesssim 0.1$ GK, owing to that the transition between ground states is
calculated to be stronger (a little weaker) than the data as seen from Fig. 3.
Since the transitions from two related low-lying states are predicted by the
PSM to be weaker than the TY87 results, at the $s$-process typical temperature
$T\approx 0.3$ GK, the predicted rate by the PSM calculations is much reduced
than the corresponding TY87 rate by about a factor of five (1.5 orders of
magnitude) without (with) the quenching of nuclear matrix elements. The
stellar $\beta^{-}$-decay rate of 204Tl is then probably much lower than what
is expected before, and the effect of the quenching on the decay rate could be
as large as about one order of magnitude. It is worth mentioning that with the
experimental developments of neutron-capture cross sections, uncertainty in
stellar $\beta$-decay rates need to be reduced with reliable nuclear many-body
models, when first-forbidden transition play important roles, the
corresponding quenching problem needs further investigation.

Finally, the partial decay rates from the three individual transitions in Fig.
3, i.e., the $\lambda^{\beta^{-}}_{if}$ weighted by corresponding thermal
population probability of initial states as shown in Eq. (1), by the PSM
calculations with the quenching factors, are plotted in Fig. 4(b) along with
the total one. At low temperature, the total rate is dominated by the
transition between ground states as the parent nucleus stays in its ground
states. The partial decay rate for transition between ground states keeps
nearly constant with the increase of $T$, indicating that only tiny population
probability is shared by the ground state to low-lying state for 204Tl at
$s$-process environments. At $T\approx 0.5$ GK, only about $3\%$ population
probability is shared, which is dominated by the $1^{-}\rightarrow 0^{+}$
transition (about $2.3\%$) due to the large degeneracy ($2J_{i}+1$) of the
$1^{-}$ state compared with the $0^{-}$ state. Besides, such transition is
predicted to be stronger than the $0^{-}\rightarrow 0^{+}$ transition, so that
the total decay rate is dominated by the $1^{-}\rightarrow 0^{+}$ transition
at high temperature, as seen from Fig. 4(b).

In summary, the first-forbidden transitions in nuclear $\beta$ decays are
expected to be important for many astrophysical problems regarding stellar
nucleosynthesis, such as the $s$ process [12], $r$ process [42] etc. In this
work we propose a practical model to calculate the stellar $\beta$-decay rates
in stellar environments, where contributions from both allowed and first-
forbidden transitions are taken into account. The model is based on the recent
developments of the projected shell model (PSM) on nuclear first-forbidden
transitions in terrestrial case and on Gamow-Teller transitions for stellar
weak-interaction rates. An important $s$-process branching-point nucleus,
204Tl, is taken as the first example, where all related transitions are first-
forbidden. In the terrestrial case, the parent nucleus stays in its ground
state, the transition from the ground state is unique first-forbidden and is
described reasonably by our calculations. With the increase of the $s$-process
temperature, although the population probability is tiny, transitions from two
low-lying states, which are non-unique first-forbidden, become relevant and
are predicted to be stronger than the transition from the ground state by five
orders of magnitude from the widely used data tables by Takahashi and Yokoi
(TY87), leading to the stellar $\beta$-decay rate of 204Tl increases very
rapidly and making 204Tl as an $s$-process thermometer and affecting the
potential chronometers of the last $s$-process events, 205Pb. The transitions
from two low-lying states are expected to be much reduced from our PSM
calculations, and our calculated stellar $\beta$-decay rate of 204Tl is much
lower than the widely used TY87 results by about a factor of five (1.5 orders
of magnitude) when no quenching factors (appropriate quenching factors) of
nuclear matrix elements are adopted, at the $s$-process typical temperature
$T\approx 0.3$ GK. Besides, the effect of the quenching on the decay rate of
204Tl is found to be as large as about one order of magnitude.

With the experimental developments of neutron-capture cross sections over the
years and in the near future, uncertainty in stellar $\beta$-decay rates need
to be reduced with reliable nuclear many-body models for understanding the $s$
process, $r$ process etc. when first-forbidden transition play important
roles, the corresponding quenching problem needs further investigation.

## Acknowledgements

L.J.W. would like to thank B. S. Gao for discussions about charge-exchange
reactions and forbidden transitions. This work is supported by the National
Natural Science Foundation of China (Grants No. 12275225), by the Fundamental
Research Funds for the Central Universities (Grant No. SWUKT22050), and
partially supported by the Key Laboratory of Nuclear Data (China Institute of
Atomic Energy).

## Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## References

  * [1] G. M. Fuller, W. A. Fowler, M. J. Newman, Stellar weak-interaction rates for sd-shell nuclei. i. nuclear matrix element systematics with application to 26Al and selected nuclei of imprtance to the supernova problem, Astrophys. J. (Suppl.) 42 (3) (1980) 447–473.  doi:Doi10.1086/190657.
URL <GotoISI>://WOS:A1980JS83500003

  * [2] G. M. Fuller, W. A. Fowler, M. J. Newman, Stellar weak-interaction rates for intermediate-mass nuclei. ii., Astrophys. J. 252 (1982) 715.
  * [3] G. M. Fuller, W. A. Fowler, M. J. Newman, Stellar weak-interaction rates for intermediate-mass nuclei. iii., Astrophys. J. (Suppl.) 48 (1982) 279.
  * [4] G. M. Fuller, W. A. Fowler, M. J. Newman, Stellar weak-interaction rates for intermediate-mass nuclei. iv., Astrophys. J. 293 (1985) 1.
  * [5] K. Langanke, G. Martínez-Pinedo, Nuclear weak-interaction processes in stars, Rev. Mod. Phys. 75 (2003) 819–862.  doi:10.1103/RevModPhys.75.819.
URL https://link.aps.org/doi/10.1103/RevModPhys.75.819

  * [6] K. Langanke, G. Martínez-Pinedo, R. G. T. Zegers, Electron capture in stars, Reports on Progress in Physics 84 (6) (2021) 066301.  doi:10.1088/1361-6633/abf207.
URL https://dx.doi.org/10.1088/1361-6633/abf207

  * [7] F. Käppeler, R. Gallino, S. Bisterzo, W. Aoki, The $s$ process: Nuclear physics, stellar models, and observations, Rev. Mod. Phys. 83 (2011) 157–193.  doi:10.1103/RevModPhys.83.157.
URL https://link.aps.org/doi/10.1103/RevModPhys.83.157

  * [8] J. J. Cowan, C. Sneden, J. E. Lawler, A. Aprahamian, M. Wiescher, K. Langanke, G. Martínez-Pinedo, F.-K. Thielemann, Origin of the heaviest elements: The rapid neutron-capture process, Rev. Mod. Phys. 93 (2021) 015002.  doi:10.1103/RevModPhys.93.015002.
URL https://link.aps.org/doi/10.1103/RevModPhys.93.015002

  * [9] H. Schatz, A. Aprahamian, J. Görres, M. Wiescher, T. Rauscher, J. Rembges, F.-K. Thielemann, B. Pfeiffer, P. Möller, K.-L. Kratz, H. Herndl, B. Brown, H. Rebel, rp-process nucleosynthesis at extreme temperature and density conditions, Physics Reports 294 (4) (1998) 167–263.  doi:https://doi.org/10.1016/S0370-1573(97)00048-3.
URL https://www.sciencedirect.com/science/article/pii/S0370157397000483

  * [10] H. Schatz, S. Gupta, P. Möller, M. Beard, E. F. Brown, A. T. Deibel, L. R. Gasques, W. R. Hix, L. Keek, R. Lau, A. W. Steiner, M. Wiescher, Strong neutrino cooling by cycles of electron capture and $\beta^{-}$ decay in neutron star crusts, Nature 505 (7481) (2014) 62\.  doi:10.1038/nature12757.
URL <GotoISI>://WOS:000329163300023

  * [11] R. Diehl, Astrophysics with radioactive Isotopes, Springer, 2018.
  * [12] K. Takahashi, K. Yokoi, Beta-decay rates of highly ionized heavy atoms in stellar interiors, Atomic Data and Nuclear Data Tables 36 (3) (1987) 375–409.  doi:https://doi.org/10.1016/0092-640X(87)90010-6.
URL https://www.sciencedirect.com/science/article/pii/0092640X87900106

  * [13] C. Domingo-Pardo, U. Abbondanno, G. Aerts, H. Álvarez-Pol, F. Alvarez-Velarde, S. Andriamonje, J. Andrzejewski, P. Assimakopoulos, L. Audouin, G. Badurek, P. Baumann, F. Bečvář, E. Berthoumieux, S. Bisterzo, F. Calviño, D. Cano-Ott, R. Capote, C. Carrapiço, P. Cennini, V. Chepel, E. Chiaveri, N. Colonna, G. Cortes, A. Couture, J. Cox, M. Dahlfors, S. David, I. Dillmann, R. Dolfini, W. Dridi, I. Duran, C. Eleftheriadis, M. Embid-Segura, L. Ferrant, A. Ferrari, R. Ferreira-Marques, L. Fitzpatrick, H. Frais-Koelbl, K. Fujii, W. Furman, R. Gallino, I. Goncalves, E. Gonzalez-Romero, A. Goverdovski, F. Gramegna, E. Griesmayer, C. Guerrero, F. Gunsing, B. Haas, R. Haight, M. Heil, A. Herrera-Martinez, M. Igashira, S. Isaev, E. Jericha, Y. Kadi, F. Käppeler, D. Karamanis, D. Karadimos, M. Kerveno, V. Ketlerov, P. Koehler, V. Konovalov, E. Kossionides, M. Krtička, C. Lamboudis, H. Leeb, A. Lindote, I. Lopes, M. Lozano, S. Lukic, J. Marganiec, S. Marrone, P. Mastinu, A. Mengoni, P. M. Milazzo, C. Moreau, M. Mosconi, F. Neves, H. Oberhummer, M. Oshima, S. O’Brien, J. Pancin, C. Papachristodoulou, C. Papadopoulos, C. Paradela, N. Patronis, A. Pavlik, P. Pavlopoulos, L. Perrot, R. Plag, A. Plompen, A. Plukis, A. Poch, C. Pretel, J. Quesada, T. Rauscher, R. Reifarth, M. Rosetti, C. Rubbia, G. Rudolf, P. Rullhusen, J. Salgado, L. Sarchiapone, I. Savvidis, C. Stephan, G. Tagliente, J. L. Tain, L. Tassan-Got, L. Tavora, R. Terlizzi, G. Vannini, P. Vaz, A. Ventura, D. Villamarin, M. C. Vincente, V. Vlachoudis, R. Vlastou, F. Voss, S. Walter, H. Wendler, M. Wiescher, K. Wisshak, Measurement of the neutron capture cross section of the $s$-only isotope ${}^{204}\mathrm{Pb}$ from 1 ev to 440 kev, Phys. Rev. C 75 (2007) 015806.  doi:10.1103/PhysRevC.75.015806.
URL https://link.aps.org/doi/10.1103/PhysRevC.75.015806

  * [14] U. Ratzel, C. Arlandini, F. Käppeler, A. Couture, M. Wiescher, R. Reifarth, R. Gallino, A. Mengoni, C. Travaglio, Nucleosynthesis at the termination point of the $s$ process, Phys. Rev. C 70 (2004) 065803.  doi:10.1103/PhysRevC.70.065803.
URL https://link.aps.org/doi/10.1103/PhysRevC.70.065803

  * [15] R. Broda, K. H. Maier, B. Fornal, J. Wrzesiński, B. Szpak, M. P. Carpenter, R. V. F. Janssens, W. Królas, T. Pawłat, S. Zhu, High-spin states and isomers in the one-proton-hole and three-neutron-hole 204tl isotope, Phys. Rev. C 84 (2011) 014330.  doi:10.1103/PhysRevC.84.014330.
URL https://link.aps.org/doi/10.1103/PhysRevC.84.014330

  * [16] H. Utsunomiya, T. Renstrøm, G. M. Tveten, S. Goriely, T. Ari-izumi, D. Filipescu, J. Kaur, Y.-W. Lui, W. Luo, S. Miyamoto, A. C. Larsen, S. Hilaire, S. Péru, A. J. Koning, $\gamma$-ray strength function for thallium isotopes relevant to the ${}^{205}\mathrm{Pb}-^{205}\mathrm{Tl}$ chronometry, Phys. Rev. C 99 (2019) 024609.  doi:10.1103/PhysRevC.99.024609.
URL https://link.aps.org/doi/10.1103/PhysRevC.99.024609

  * [17] A. Casanovas, A. E. Tarifeño-Saldivia, C. Domingo-Pardo, et al., Neutron capture measurement at the n tof facility of the 204tl and 205tl s-process branching points, Journal of Physics: Conference Series 1668 (1) (2020) 012005\.  doi:10.1088/1742-6596/1668/1/012005.
URL https://dx.doi.org/10.1088/1742-6596/1668/1/012005

  * [18] A. A. Casanovas, C. Domingo-Pardo, C. Guerrero, et al., Measurement of the radiative capture cross section of the s-process branching points 204tl and 171tm at the n tof facility (cern), EPJ Web of Conferences 178 (2018) 03004.  doi:10.1051/epjconf/201817803004.
URL https://doi.org/10.1051/epjconf/201817803004

  * [19] A. Casanovas-Hoste, C. Domingo-Pardo, C. Guerrero, et al., Analysis of the impact of the 204tl neutron capture cross section on the s-process only isotope 204pb, EPJ Web of Conferences 260 (2022) 02002.  doi:10.1051/epjconf/202226002002.
URL https://doi.org/10.1051/epjconf/202226002002

  * [20] B. Gao, S. Giraud, K. A. Li, A. Sieverding, R. G. T. Zegers, X. Tang, J. Ash, Y. Ayyad-Limonge, D. Bazin, S. Biswas, B. A. Brown, J. Chen, M. DeNudt, P. Farris, J. M. Gabler, A. Gade, T. Ginter, M. Grinder, A. Heger, C. Hultquist, A. M. Hill, H. Iwasaki, E. Kwan, J. Li, B. Longfellow, C. Maher, F. Ndayisabye, S. Noji, J. Pereira, C. Qi, J. Rebenstock, A. Revel, D. Rhodes, A. Sanchez, J. Schmitt, C. Sumithrarachchi, B. H. Sun, D. Weisshaar, New ${}^{59}\mathrm{Fe}$ stellar decay rate with implications for the ${}^{60}\mathrm{Fe}$ radioactivity in massive stars, Phys. Rev. Lett. 126 (2021) 152701.  doi:10.1103/PhysRevLett.126.152701.
URL https://link.aps.org/doi/10.1103/PhysRevLett.126.152701

  * [21] K.-A. Li, C. Qi, M. Lugaro, A. Y. López, A. I. Karakas, J. den Hartogh, B.-S. Gao, X.-D. Tang, The stellar $\beta$-decay rate of 134cs and its impact on the barium nucleosynthesis in the s-process, The Astrophysical Journal Letters 919 (2) (2021) L19.  doi:10.3847/2041-8213/ac260f.
URL https://dx.doi.org/10.3847/2041-8213/ac260f

  * [22] K. Hara, Y. Sun, Projected shell model and high-spin spectroscopy, Int. J. Mod. Phys. E 4 (04) (1995) 637–785.
  * [23] Y. Sun, D. H. Feng, High spin spectroscopy with the projected shell model, Phys. Rep. 264 (1996) 375.  doi:https://doi.org/10.1016/0370-1573(95)00049-6.
URL http://www.sciencedirect.com/science/article/pii/0370157395000496

  * [24] T. Mizusaki, M. Oi, F.-Q. Chen, Y. Sun, Grassmann integral and Balian–Brézin decomposition in Hartree-Fock-Bogoliubov matrix elements, Phys. Lett. B 725 (1) (2013) 175–179.  doi:https://doi.org/10.1016/j.physletb.2013.07.005.
URL https://www.sciencedirect.com/science/article/pii/S0370269313005558

  * [25] L.-J. Wang, F.-Q. Chen, T. Mizusaki, M. Oi, Y. Sun, Toward extremes of angular momentum: Application of the pfaffian algorithm in realistic calculations, Phys. Rev. C 90 (2014) 011303(R).  doi:10.1103/PhysRevC.90.011303.
URL https://link.aps.org/doi/10.1103/PhysRevC.90.011303

  * [26] L.-J. Wang, Y. Sun, T. Mizusaki, M. Oi, S. K. Ghorui, Reduction of collectivity at very high spins in ${}^{134}\mathrm{Nd}$: Expanding the projected-shell-model basis up to 10-quasiparticle states, Phys. Rev. C 93 (2016) 034322.  doi:10.1103/PhysRevC.93.034322.
URL https://link.aps.org/doi/10.1103/PhysRevC.93.034322

  * [27] Z.-R. Chen, L.-J. Wang, Pfaffian formulation for matrix elements of three-body operators in multiple quasiparticle configurations, Phys. Rev. C 105 (2022) 034342.  doi:10.1103/PhysRevC.105.034342.
URL https://link.aps.org/doi/10.1103/PhysRevC.105.034342

  * [28] B.-L. Wang, F. Gao, L.-J. Wang, Y. Sun, Effective and efficient algorithm for the wigner rotation matrix at high angular momenta, Phys. Rev. C 106 (2022) 054320.  doi:10.1103/PhysRevC.106.054320.
URL https://link.aps.org/doi/10.1103/PhysRevC.106.054320

  * [29] L.-J. Wang, J. Dong, F.-Q. Chen, Y. Sun, Projected shell model analysis of structural evolution and chaoticity in fast-rotating nuclei, Journal of Physics G: Nuclear and Particle Physics 46 (10) (2019) 105102.  doi:10.1088/1361-6471/ab33be.
URL https://dx.doi.org/10.1088/1361-6471/ab33be

  * [30] L.-J. Wang, F.-Q. Chen, Y. Sun, Basis-dependent measures and analysis uncertainties in nuclear chaoticity, Phys. Lett. B 808 (2020) 135676.  doi:https://doi.org/10.1016/j.physletb.2020.135676.
URL https://www.sciencedirect.com/science/article/pii/S0370269320304792

  * [31] Z.-C. Gao, Y. Sun, Y.-S. Chen, Shell model method for gamow-teller transitions in heavy, deformed nuclei, Phys. Rev. C 74 (2006) 054303.  doi:10.1103/PhysRevC.74.054303.
URL https://link.aps.org/doi/10.1103/PhysRevC.74.054303

  * [32] L.-J. Wang, Y. Sun, S. K. Ghorui, Shell-model method for gamow-teller transitions in heavy deformed odd-mass nuclei, Phys. Rev. C 97 (2018) 044302.  doi:10.1103/PhysRevC.97.044302.
URL https://link.aps.org/doi/10.1103/PhysRevC.97.044302

  * [33] L. Tan, Y.-X. Liu, L.-J. Wang, Z. Li, Y. Sun, A novel method for stellar electron-capture rates of excited nuclear states, Phys. Lett. B 805 (2020) 135432.  doi:https://doi.org/10.1016/j.physletb.2020.135432.
URL https://www.sciencedirect.com/science/article/pii/S0370269320302367

  * [34] L.-J. Wang, L. Tan, Z. Li, G. W. Misch, Y. Sun, Urca cooling in neutron star crusts and oceans: Effects of nuclear excitations, Phys. Rev. Lett. 127 (2021) 172702.  doi:10.1103/PhysRevLett.127.172702.
URL https://link.aps.org/doi/10.1103/PhysRevLett.127.172702

  * [35] L.-J. Wang, L. Tan, Z. Li, B. Gao, Y. Sun, Description of ${}^{93}\mathrm{Nb}$ stellar electron-capture rates by the projected shell model, Phys. Rev. C 104 (2021) 064323.  doi:10.1103/PhysRevC.104.064323.
URL https://link.aps.org/doi/10.1103/PhysRevC.104.064323

  * [36] Z.-R. Chen, L.-J. Wang, Stellar $\beta$\- decay rates for ${}^{63}\text{Co}$ and ${}^{63}\text{Ni}$ by the projected shell model, Symmetry 15 (2) (2023) 315.  doi:10.3390/sym15020315.
URL https://www.mdpi.com/2073-8994/15/2/315

  * [37] Z.-R. Chen, L.-J. Wang, Stellar weak-interaction rates for $rp$-process waiting-point nuclei from projected shell model (2023 Phys. Lett. B in press).  arXiv:2308.11297.
  * [38] F. Gao, Z.-R. Chen, L.-J. Wang, Nuclear $\beta$ spectrum from projected shell model (i): allowed one-to-one transition (2023 Phys. Rev. C in press).  arXiv:2308.11299.
  * [39] B.-L. Wang, L.-J. Wang, First-forbidden transition of nuclear $\beta$ decay by projected shell model (2023).  arXiv:2310.19523.
  * [40] J. C. Hardy, I. S. Towner, Superallowed ${0}^{+}\rightarrow{0}^{+}$ nuclear $\beta$ decays: A new survey with precision tests of the conserved vector current hypothesis and the standard model, Phys. Rev. C 79 (2009) 055502.  doi:10.1103/PhysRevC.79.055502.
URL https://link.aps.org/doi/10.1103/PhysRevC.79.055502

  * [41] G. K. Schenter, P. Vogel, A simple approximation of the fermi function in nuclear beta decay, Nuclear Science and Engineering 83 (3) (1983) 393–396.  arXiv:https://doi.org/10.13182/NSE83-A17574, doi:10.13182/NSE83-A17574.
URL https://doi.org/10.13182/NSE83-A17574

  * [42] Q. Zhi, E. Caurier, J. J. Cuenca-García, K. Langanke, G. Martínez-Pinedo, K. Sieja, Shell-model half-lives including first-forbidden contributions for $r$-process waiting-point nuclei, Phys. Rev. C 87 (2013) 025803.  doi:10.1103/PhysRevC.87.025803.
URL https://link.aps.org/doi/10.1103/PhysRevC.87.025803

  * [43] A. L. Cole, T. S. Anderson, R. G. T. Zegers, S. M. Austin, B. A. Brown, L. Valdez, S. Gupta, G. W. Hitt, O. Fawwaz, Gamow-teller strengths and electron-capture rates for $pf$-shell nuclei of relevance for late stellar evolution, Phys. Rev. C 86 (2012) 015809.  doi:10.1103/PhysRevC.86.015809.
URL https://link.aps.org/doi/10.1103/PhysRevC.86.015809

  * [44] P. Sarriguren, Stellar electron-capture rates in $pf$-shell nuclei from quasiparticle random-phase approximation calculations, Phys. Rev. C 87 (2013) 045801.  doi:10.1103/PhysRevC.87.045801.
URL https://link.aps.org/doi/10.1103/PhysRevC.87.045801

  * [45] G. Martínez-Pinedo, Y. H. Lam, K. Langanke, R. G. T. Zegers, C. Sullivan, Astrophysical weak-interaction rates for selected $a=20$ and $a=24$ nuclei, Phys. Rev. C 89 (2014) 045806.  doi:10.1103/PhysRevC.89.045806.
URL https://link.aps.org/doi/10.1103/PhysRevC.89.045806

  * [46] B. A. Brown, B. H. Wildenthal, Experimental and theoretical gamow-teller beta-decay observables for the sd-shell nuclei, Atomic Data and Nuclear Data Tables 33 (3) (1985) 347.
  * [47] G. Martínez-Pinedo, A. Poves, E. Caurier, A. P. Zuker, Effective ${g}_{A}$ in the $\mathrm{pf}$ shell, Phys. Rev. C 53 (1996) R2602–R2605.  doi:10.1103/PhysRevC.53.R2602.
URL https://link.aps.org/doi/10.1103/PhysRevC.53.R2602

  * [48] J. Menéndez, D. Gazit, A. Schwenk, Chiral two-body currents in nuclei: Gamow-teller transitions and neutrinoless double-beta decay, Phys. Rev. Lett. 107 (2011) 062501.  doi:10.1103/PhysRevLett.107.062501.
URL https://link.aps.org/doi/10.1103/PhysRevLett.107.062501

  * [49] L.-J. Wang, J. Engel, J. M. Yao, Quenching of nuclear matrix elements for $0\nu\beta\beta$ decay by chiral two-body currents, Phys. Rev. C 98 (2018) 031301(R).  doi:10.1103/PhysRevC.98.031301.
URL https://link.aps.org/doi/10.1103/PhysRevC.98.031301

  * [50] B. Märkisch, H. Mest, H. Saul, X. Wang, H. Abele, D. Dubbers, M. Klopf, A. Petoukhov, C. Roick, T. Soldner, D. Werder, Measurement of the weak axial-vector coupling constant in the decay of free neutrons using a pulsed cold neutron beam, Phys. Rev. Lett. 122 (2019) 242501.  doi:10.1103/PhysRevLett.122.242501.
URL https://link.aps.org/doi/10.1103/PhysRevLett.122.242501

  * [51] H. A. Weidenmüller, First-forbidden beta decay, Rev. Mod. Phys. 33 (1961) 574–607.  doi:10.1103/RevModPhys.33.574.
URL https://link.aps.org/doi/10.1103/RevModPhys.33.574

  * [52] X. Mougeot, Reliability of usual assumptions in the calculation of $\beta$ and $\nu$ spectra, Phys. Rev. C 91 (2015) 055504.  doi:10.1103/PhysRevC.91.055504.
URL https://link.aps.org/doi/10.1103/PhysRevC.91.055504

  * [53] M. Haaranen, J. Kotila, J. Suhonen, Spectrum-shape method and the next-to-leading-order terms of the $\beta$-decay shape factor, Phys. Rev. C 95 (2017) 024327.  doi:10.1103/PhysRevC.95.024327.
URL https://link.aps.org/doi/10.1103/PhysRevC.95.024327

  * [54] https://www.nndc.bnl.gov.