# Weyl Geometry and Quantum Corrections

Sijo K. Joseph  Department of Physics, GITAM (Deemed to be University),
Hyderabad, India<EMAIL_ADDRESS>
###### Abstract.

Recent research in the geometric formulation of quantum theory has implied
that Weyl Geometry can be used to merge quantum theory and general relativity
consistently as classical field theories. In the Weyl Geometric framework, it
seems that both quantum theory and gravity can merge consistently, once
quantum theory is geometrized. The extended differential geometry can modify
the quantum mechanical results into a more general nonlinear framework. Author
shows that, how the extended differential geometry modifies the known quantum
equations and also the modification to the Maxwell’s electromagnetic
equations.

###### Key words and phrases:

Weyl Geometry, Quantum Theory, Classical Electromagnetism

###### 1991 Mathematics Subject Classification:

53A99, 81-02, 78-06

##  1\. Introduction

It is well know that Einstein’s theory of gravity is based on Riemann Geometry
which is only a subclass of differential geometry. There are many mathematical
extensions to Riemann geometry, for example the Weyl Geometry is one among
them [WeylReview2017]. Einstein’s gravity assumes a simple Lagrangian
proportional to the curvature scalar ($\mathcal{L}=\sqrt{-g}R$ ). This is one
of the simplest Lagrangians that one can assume. There are many possible
generalizations of Einstein’s theory, typical examples can be seen in these
references [BeyondEinsteinGravity, Bergmann1968, fR_HamFormlation, fofT1,
fofT2, Bekenstein2011_TeVeS, BekensteinPRD_TeVeS, Moffat_STVG, MOG_Moffat].

Weyl had developed a purely infinitesimal geometry, building upon a conformal
generalization of the Riemannian metric $g_{\mu\nu}$, where a point-dependent
rescaling is acheived via the the conformal factor $\Omega^{2}$ and it is
given by $g_{\mu\nu}=\Omega^{2}\bar{g}_{\mu\nu}$ where $\bar{g}_{\mu\nu}$ is
our background metric. It was F. Shojai et.al. [Shojai2008] who discovered
that the quantum theory can be incorporated into gravity in a geometric manner
using the conformal factor $\Omega^{2}$. Conformal re-scaling arises purely
from a quantum mechanical quantity called the quantum potential
[Shojai_Article, GabayJoseph1, GabayJoseph2, SKJoseph1]. Such a geometrical
unification of quantum theory to general theory of relativity is achieved
using deBroglie-Bohm version of quantum theory [Bohm1975, BohmI, BohmII,
Shojai_Weyl2003]. Author is reviewing the Weyl Geometry and the correction
terms appearing in scalar quantum mechanical wave-equation based on the
previous studies [GabayJoseph1, GabayJoseph2, Joseph2018Geom]. We will simply
focus on the mathematical corrections appearing in quantum mechanical
equations as well as in the Maxwell’s electromagnetic equations.

##  2\. Weyl Gravity Corrections in Klein-Gordon Equation

In Riemannian geometry, the length of a vector remains constant and only the
orientation changes during parallel transport . While in Weyl Geometry, both
the length of a vector and its orientation changes during parallel transport.

(2.1) $\displaystyle\nabla_{\alpha}g_{\mu\nu}=0$ $\displaystyle\implies$
Riemannian Geometry (2.2)
$\displaystyle\nabla_{\alpha}g_{\mu\nu}=\sigma_{\alpha}g_{\mu\nu}$
$\displaystyle\implies$ Weyl Geometry

In Riemannian geometry the covariant derivative of the metric is zero while it
is not true in Weyl Geometry. Since the length changes during the parallel
transport, the covariant derivative of the metric gives a nonzero
contribution.

Weyl geometry encountered a serious criticism by Einstien, he criticized that
the clock will tick based on its history and different clock with different
history will click at different rates which is not observed in reality hence
Weyl theory cannot be correct. That is a criticism posed by Einstein in order
to reject Weyl Geometry. This criticism can be easily overcome by adopting the
Integrable Weyl Geometry. Weyl integrable geometry is given by the following
condition,

(2.3) $\sigma=d\phi,$

here $\phi$ is a scalar field. Once $\sigma$ is a perfect differential of a
scalar field, the integral over a closed curve vanishes. Using this argument,
the criticism by Einstein can be circumvented, if the clock is completed a
closed route, then the integral vanishes, and the integral is no-longer path
dependent. Hence in integrable Weyl geomety, Einstein’s criticism is no longer
valid. In Weyl geometry, one can think about different conformal frames where
the metric is different by a conformal scaling but the Physics remains the
sames.

Here the following equation i.e. Eq. 2.4 gives a different description of the
same manifold with a conformally re-scaled metric and Weyl field $\sigma$.

(2.4) $(M,g,\sigma)\to(M,\bar{g},\bar{\sigma}).$

The different conformal frames are related by the following transformation
equations and it is called Weyl Gauge transformation,

(2.5) $\displaystyle\bar{g}={e}^{-f}g$ (2.6)
$\displaystyle\bar{\sigma}=\sigma-df$

where $f$ is a scalar function on the manifold $M$. Let us concentrate on the
conformal transformation of the metric in different Weyl frames.

(2.7) $(M,g,\sigma)\to(M,\bar{g},\bar{\sigma})$ (2.8)
$\displaystyle\bar{g}={e}^{-f}g$

Different Weyl frames are related via conformal transformations yet it
describe the same Physics. This concept can be easily understood, once we
focus on the geometric formulation of quantum theory using conformal
transformation.

##  3\. Geometric Formulation of Quantum Theory

Here our aim is to show that quantum theory is associated to the conformal
frames as described in Weyl Geometry. Consider the Lagrangian density of
Klein-Gordon Field, which is given by,

(3.1)
$\displaystyle\mathcal{L}=\frac{1}{2}\partial_{\mu}\Phi^{*}\partial^{\mu}\Phi-\frac{1}{2}\frac{m^{2}}{\hbar^{2}}\Phi^{*}\Phi$

Euler-Lagrange equation gives,

(3.2) $\displaystyle\mathcal{S}=\int{\mathcal{L}\sqrt{-g}\,\mathrm{d}^{4}x}.$
(3.3)
$\displaystyle\frac{\delta\mathcal{S}}{\delta\phi}=0\implies\frac{\partial\mathcal{L}}{\partial\phi}-\partial_{\mu}\left(\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\phi)}\right)=0$

Hence the Klein-Gordon Equation is obtained using standard quantum theory.

(3.4)
$\displaystyle(\partial_{\mu}\partial^{\mu}+\frac{m^{2}}{\hbar^{2}})\Phi=0$

Note that, once can use a complex field to describe the quantum theory.
Gravity has a highly different mathematical framework which uses real
quantities and differential geometry is utilized to describe the physical
mechanism. In quantum theory, we adopt complex fields on a flat Minkowski
space-time. In order to geometrize quantum theory, we need to return to the
real field variables, hence decomposing the complex field into two real field
equation, one can geometrize quantum theory. Taking
$\Phi=\sqrt{\rho}{e}^{\frac{i}{\hbar}S}$, we will be able to decompose the
Klein-Gordon equation into two real field equations. Substituting polar form
of $\Phi$ in Eq. 3.4 and separating real and imaginary part we get two
different field equations.

(3.5) $\displaystyle\partial_{\mu}S\partial^{\mu}S$ $\displaystyle=$
$\displaystyle m^{2}\Omega^{2}$ (3.6)
$\displaystyle\partial_{\mu}(\rho\partial^{\mu}S)$ $\displaystyle=$
$\displaystyle 0$

where
$\Omega^{2}=1+\frac{\hbar^{2}}{m^{2}}\frac{\partial_{\mu}\partial^{\mu}\sqrt{\rho}}{\sqrt{\rho}}$.
One can notice that $\Omega^{2}$ looks like a conformal factor here. If we
start with classical equations and make a conformal transformation, one can
get the quantum mechanical equations. This is a differential geometric way of
finding quantum equations. Once can see that with a proper conformal scaling
of the Minkowski space, one can reach into classical equation of motion. For
example, Eq. 3.5, can be re-casted into differential geometric form by
dividing it with $\Omega^{2}$.

(3.7)
$\displaystyle\frac{1}{\Omega^{2}}\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}S$
$\displaystyle=$ $\displaystyle m^{2}$ (3.8) $\displaystyle
g^{\mu\nu}\partial_{\mu}S\partial_{\nu}S$ $\displaystyle=$ $\displaystyle
m^{2}$

Here the new metric $g_{\mu\nu}$ can be written as a conformal transformation
to the Minkowski metric ${\eta_{\mu\nu}}$.

(3.9) $\displaystyle g^{\mu\nu}=\frac{\eta^{\mu\nu}}{\Omega^{2}}\implies
g_{\mu\nu}={\Omega^{2}}{\eta_{\mu\nu}}$

Here Eq. 3.8 obtained looks like a free-particle classical equation motion on
a curved manifold with metric $g_{\mu\nu}={\Omega^{2}}{\eta_{\mu\nu}}$. Hence
a conformal re-scaling of Minkowski space-time $\eta_{\mu\nu}$ makes the
quantum equations into classical equations of motion. In other words, quantum
theory can be absorbed into a conformally transformed metric. Treating the
space-time as conformally curved and exploring the classical motion on this
manifold is theoretically equivalent to studying a quantum problem on a flat
Minkowski space-time. This opens the freedom to choose different conformal
frames. For example when the background metric is curved due to gravity ie
$\bar{g}_{\mu\nu}$, we can consider a re-scaled metric
${g}_{\mu\nu}=\Omega^{2}\bar{g}_{\mu\nu}$. There are many ways to choose this
total metric ${g}_{\mu\nu}$, hence there are different conformal frames which
can give same Physics as long as the total metric is the same. These conformal
frames corresponds to different ways of mixing quantum theory and Einstein
gravity. Hence incorporating quantum theory in differential geometric
framework one is naturally compelled to extend differential geometry in a much
wider mathematical framework. One can associate gravity to the the coordinate
transformation while quantum theory is associated with the length change
during parallel transport of a vector. Hence Weyl Geometry is a mathematical
framework to hold both quantum theory and Einstein’s gravity.

##  4\. Coupling Gravity with Geometrized Quantum Theory

Let us begin with the following classical action,

(4.1) $\displaystyle A[g_{\mu\nu},S,\rho,\lambda]$ $\displaystyle=$
$\displaystyle{\frac{1}{2\kappa}\int{d^{4}x\sqrt{-g}R}}$
$\displaystyle{+\int{d^{4}x\sqrt{-g}\left(\frac{\rho}{m}\nabla_{\mu}S\nabla^{\mu}S-m\rho\right)}}$

In order to incorporate the quantum effect, one need to make a conformal
transformation to the aforementioned action and one can get,

$\displaystyle
A[g_{\mu\nu},{\Omega^{2}},S,\rho,\lambda]={\frac{1}{2\kappa}\int{d^{4}x\sqrt{-g}\left(R\Omega^{2}-6\nabla_{\mu}\Omega\nabla^{\mu}\Omega\right)}}$
$\displaystyle{+\int{d^{4}x\sqrt{-g}\left(\frac{\rho}{m}\Omega^{2}\nabla_{\mu}S\nabla^{\mu}S-m\rho\Omega^{4}\right)}}$
(4.2)
$\displaystyle{+\int{d^{4}x\sqrt{-g}\lambda\left[\ln{\Omega^{2}}-\left(\frac{\hbar^{2}}{m^{2}}\frac{\nabla_{\mu}\nabla^{\mu}\sqrt{\rho}}{\sqrt{\rho}}\right)\right]}}.$

In order to get correct quantum mechanical equations, one needs to impose a
constraint condition which is done in the third term using a Lagrange
multiplier $\lambda$. The constraint equation is given by,

(4.3)
$\displaystyle{\Omega^{2}=\exp\Bigl{(}\frac{\hbar^{2}}{m^{2}}\frac{\nabla_{\mu}\nabla^{\mu}\sqrt{\rho}}{\sqrt{\rho}}}\Bigr{)}\approx
1+\frac{\hbar^{2}}{m^{2}}\frac{\nabla_{\mu}\nabla^{\mu}\sqrt{\rho}}{\sqrt{\rho}}$

Minimizing the action, one is lead to generalized equations for the real-
fields $S$ and $\rho$ and it is given by,

(4.4)
$\displaystyle{(\nabla_{\mu}S\nabla^{\mu}S-m^{2}\Omega^{2})+\frac{\hbar^{2}}{2m\Omega^{2}\sqrt{\rho}}[\Box({\frac{\lambda}{\sqrt{\rho}})}-\lambda\frac{\Box\sqrt{\rho}}{\rho}]=0}$
(4.5) $\displaystyle{\nabla_{\nu}(\rho\Omega^{2}\nabla^{\nu}S)=0}$

Here Eq. 4.4 and Eq. 4.5 together gives a generalized Klein-Gordon Equation
once seen in the wavefunction picture
$\Phi=\sqrt{\rho}{e}^{\frac{i}{\hbar}S}$. Even in the flat space-time
$\lambda$ correction exists in Eq. 4.4. Combining Eq. 4.4 and Eq. 4.5, one can
get the complex quantum equation which is a generalized Klein-Gordon equation
with source and dissipation terms.

(4.6)
$\displaystyle{\Box\Phi+\frac{i}{\hbar}\frac{\nabla_{\mu}\Omega^{2}}{\Omega^{2}}\nabla^{\mu}S\Phi+\frac{m^{2}}{\hbar^{2}}\Phi=\frac{1}{2m\Omega^{2}\sqrt{\rho}}[\Box({\frac{\lambda}{\sqrt{\rho}})}-\lambda\frac{\Box\sqrt{\rho}}{\rho}]\Phi}$

It simultaneously satisfy the following $\lambda$ equation (Eq. 4.7),

(4.7)
$\displaystyle{\lambda=\frac{1}{(1-Q)}\frac{\hbar^{2}}{m^{2}}\nabla_{\sigma}(\lambda\frac{\nabla^{\sigma}{\sqrt{\rho}}}{\sqrt{\rho}})}.$

Hence we are lead to a more general quantum theory with the presence of a
dissipation and source contribution. Taking $\lambda=0$ or $\lambda=\rho$, one
can get the complex Klein-Gordon equation without the source term.

(4.8)
$\displaystyle{\Box\Phi+\Bigl{(}\frac{i}{\hbar}\frac{\nabla_{\mu}\Omega^{2}}{\Omega^{2}}\nabla^{\mu}S\Bigr{)}\Phi+\frac{m^{2}}{\hbar^{2}}\Phi=0}$

If we assume that quantum force coupling contribution
($\Bigl{(}\frac{i}{\hbar}\frac{\nabla_{\mu}\Omega^{2}}{\Omega^{2}}\nabla^{\mu}S\Bigr{)}\Phi$
) is negligible then one can recover standard Klein-Gordon Equations.

(4.9) $\displaystyle{\Box\Phi+\frac{m^{2}}{\hbar^{2}}\Phi}=0$

##  5\. Gravitational Corrections in Maxwell’s Equations

In order to couple electro-magnetism with matter field equation, we will
simply add the Maxwell’s Lagragian and the matter current is modified
accordingly. The following action is proposed for the electromagnetic coupling
(see [Joseph2018Geom] for more details),

$\displaystyle
A[g_{\mu\nu},{\Omega},S,\rho,A_{\mu},\lambda]=\frac{1}{2k}\int{d^{4}x\sqrt{-g}\left(R\Omega^{2}-6\nabla_{\mu}\Omega\nabla^{\mu}\Omega\right)}$
$\displaystyle+\int{d^{4}x\sqrt{-g}\left(\frac{\rho}{m}\Omega^{2}(\nabla_{\mu}{S}+eA_{\mu})(\nabla^{\mu}{S}+eA^{\mu})-m\rho\Omega^{4}\right)}$
$\displaystyle-\frac{1}{4}\int{d^{4}x\sqrt{-g}\,F_{\mu\nu}F^{\mu\nu}}$ (5.1)
$\displaystyle+\int{d^{4}x\sqrt{-g}\lambda\left[\ln(\Omega^{2})-\left(\frac{\hbar^{2}}{m^{2}}\frac{\nabla_{\mu}\nabla^{\mu}\sqrt{\rho}}{\sqrt{\rho}}\right)\right]},$

Maxwell’s equation can be found by varying the action with respect to
$A_{\mu}$,

(5.2)
$\displaystyle\nabla_{\mu}F^{\mu\nu}-\frac{2e}{m}\rho\Omega^{2}(\nabla^{\nu}S+eA^{\nu})=0.$

Continuity equation is obtained from the variation of action with respect to
$S$. This is just the imaginary part of the quantum gravity corrected Klein-
Gordon-Maxwell equation.

(5.3)
$\displaystyle\nabla_{\mu}\Bigl{(}\rho\Omega^{2}(\nabla^{\mu}S+eA^{\mu})\Bigr{)}=0$

In Eq. 5.4 we get a more generalized Electromagnetic equation. Let us consider
the flat space-time and see how this equation behaves in a Minkowski space-
time, then one can get

(5.4)
$\displaystyle\partial_{\mu}F^{\mu\nu}-\frac{2e}{m}\rho\Omega^{2}(\partial^{\nu}S+eA^{\nu})=0.$

Rewriting the expression of $F^{\mu\nu}$ in terms of the electro-magnetic
vector potential $A^{\nu}$ in lorentz gauge, one can see that

(5.5)
$\displaystyle\Bigl{(}\Box-\frac{2e^{2}\rho}{m}\Omega^{2}\Bigr{)}A^{\nu}=\frac{2e}{m}\rho\Omega^{2}\partial^{\nu}S.$

Note that quantum mechanical current is simply
$J_{qm}^{\nu}=\rho\partial^{\nu}S$ while we have a conformally corrected
quantum current density $J^{\prime\nu}=\Omega^{2}\rho\partial^{\nu}S$.Once
$\Omega^{2}=e^{Q}$ is taken into account and higher order terms are neglected
the total current density can be expressed as $J^{\prime\nu}\approx
J_{qm}^{\nu}+QJ_{qm}^{\nu}$. Then the generalized Maxwells equations can be
written as

(5.6)
$\displaystyle\Bigl{(}\Box-\frac{2e^{2}\rho}{m}(1+Q)\Bigr{)}A^{\nu}=\frac{2e}{m}\Bigl{(}J_{qm}^{\nu}+QJ_{qm}^{\nu}\Bigr{)},$

where $Q$ is the quantum potential. One can see that Maxwell’s equation is
corrected with a Proca like mass term with negative signature. Similarly the
quantum mechanical current contains a contribution from the Conformal factor.
Note that the conformal factor is given by $\Omega^{2}=e^{Q}$, Eq. 5.6 is
found by ignoring higher order $\hbar$ terms, and one can see that in Eq. 5.4,
when the quantum potential $Q\to 0$ one can recover the massless Maxwell’s
equation.

##  6\. Conclusions

In this article we have seen that quantum mechanical equation contains a
dissipation and source term correction due to the coupling between gravity and
quantum theory. Similarly in electromagnetism a mass term and an extra
conformal current term appears as the gravitational correction. The coupling
is not only due to the curved background geometry, even in flat space, the
correction terms appears even though it is a small quantity. For all the
practical purposes, we can deal with the linear partial differential equation
while dealing with the quantum theory but this is only a special case of the
gravitationally corrected rich nonlinear quantum equations.

## References


$\operatorname{tr}(P^{\operatorname{T}}H(P)P)=2(1-\theta)f(P).$

Let $\hat{a}=\operatorname{tr}(\widehat{P}^{\operatorname{T}}A\widehat{P})$.
By assumption (F.1), we have

$\displaystyle\eta+2(1-\theta)f(P)$
$\displaystyle\leq\operatorname{tr}(\widehat{P}^{\operatorname{T}}H(P)\widehat{P})$
$\displaystyle\leq\frac{2}{b^{\theta}}[\hat{a}+\operatorname{tr}(\widehat{P}^{\operatorname{T}}DP^{\operatorname{T}}\widehat{P})-\theta
f_{1}(P)\hat{b}],$ $\displaystyle b^{\theta}\eta+(1-\theta)f(P)b^{\theta}$
$\displaystyle\leq\hat{a}+\operatorname{tr}(\widehat{P}^{\operatorname{T}}DP^{\operatorname{T}}\widehat{P})-\theta
f_{1}(P)\hat{b},$
$\displaystyle\frac{b^{\theta}}{\hat{b}^{\theta}}\eta+(1-\theta)f(P)\frac{b^{\theta}}{\hat{b}^{\theta}}$
$\displaystyle\leq
g(\widehat{P})+\frac{\operatorname{tr}(\widehat{P}^{\operatorname{T}}DP^{\operatorname{T}}\widehat{P})}{\hat{b}^{\theta}}-\theta
f(P)\frac{\hat{b}^{1-\theta}}{b^{1-\theta}},$

implying

$g(\widehat{P})+\frac{\operatorname{tr}(\widehat{P}^{\operatorname{T}}DP^{\operatorname{T}}\widehat{P})}{\hat{b}^{\theta}}\geq
f(P)+h+\frac{b^{\theta}}{\hat{b}^{\theta}}\eta,$

where

$\displaystyle h$
$\displaystyle=(1-\theta)f(P)\frac{b^{\theta}}{\hat{b}^{\theta}}+\theta
f(P)\frac{\hat{b}^{1-\theta}}{b^{1-\theta}}-f(P)$
$\displaystyle=(1-\theta)\frac{a+d}{\hat{b}^{\theta}}+\theta\,\frac{a+d}{b}\,\hat{b}^{1-\theta}-\frac{a+d}{b^{\theta}}$
$\displaystyle=\frac{a+d}{\hat{b}^{\theta}b}\Big{[}(1-\theta)b+\theta\hat{b}-b^{1-\theta}\hat{b}^{\theta}\Big{]}.$

This proves inequality (F.2), and it is strict if inequality (F.1) is strict.
∎

The next theorem is a refinement of [67, Theorem 2.2].

######  Theorem F.1.

For $P,\widehat{P}\in\mathbb{O}^{n\times k}$, suppose either
$\theta\in\\{0,1\\}$ or
$\operatorname{tr}(P^{\operatorname{T}}AP+P^{\operatorname{T}}D)\geq 0$, and
let $\widetilde{P}=\widehat{P}Q$ where $Q\in\mathbb{O}^{k\times k}$ is an
orthogonal polar factor of $\widehat{P}^{\operatorname{T}}D$. If (F.1) holds,
then

$\frac{b^{\theta}}{\hat{b}^{\theta}}\eta+f(P)\leq
g(\widehat{P})+\frac{\operatorname{tr}(\widehat{P}^{\operatorname{T}}DP^{\operatorname{T}}\widehat{P})}{[\operatorname{tr}(\widehat{P}^{\operatorname{T}}B\widehat{P})]^{\theta}}$
(F.4)

yielding

$\displaystyle f(\widetilde{P})$ $\displaystyle\geq
f(P)+\frac{b^{\theta}}{\hat{b}^{\theta}}\eta+\frac{\|\widehat{P}^{\operatorname{T}}D\|_{\operatorname{tr}}-\operatorname{tr}(\widehat{P}^{\operatorname{T}}DP^{\operatorname{T}}\widehat{P})}{\hat{b}^{\theta}}$
(F.5) $\displaystyle\geq
f(P)+\frac{1}{2}\left(\frac{s_{k}(B)}{S_{k}(B)}\right)^{\theta}\,\eta+[S_{k}(B)]^{-\theta}\,\Big{[}\|\widehat{P}^{\operatorname{T}}D\|_{\operatorname{tr}}-\operatorname{tr}(\widehat{P}^{\operatorname{T}}DP^{\operatorname{T}}\widehat{P})\Big{]},$
(F.6)

where $s_{k}(B)$ and $S_{k}(B)$ are the sum of the $k$ smallest eigenvalues
and that of the $k$ largest eigenvalues of $B$, respectively.

###### Proof.

In Lemma F.1, we note $h\equiv 0$ in the case $\theta\in\\{0,1\\}$, and $h\geq
0$ in the case
$a+d=\operatorname{tr}(P^{\operatorname{T}}AP+P^{\operatorname{T}}D)\geq 0$,
and hence we have (F.4) which yields (F.5) upon noticing
$\operatorname{tr}(\widehat{P}^{\operatorname{T}}A\widehat{P})=\operatorname{tr}(\widetilde{P}^{\operatorname{T}}A\widetilde{P})$,
$\operatorname{tr}(\widehat{P}^{\operatorname{T}}B\widehat{P})=\operatorname{tr}(\widetilde{P}^{\operatorname{T}}B\widetilde{P})$,
and writing

$\displaystyle\operatorname{tr}(\widehat{P}^{\operatorname{T}}DP^{\operatorname{T}}\widehat{P})$
$\displaystyle=\|\widehat{P}^{\operatorname{T}}D\|_{\operatorname{tr}}-\Big{[}\|\widehat{P}^{\operatorname{T}}D\|_{\operatorname{tr}}-\operatorname{tr}(\widehat{P}^{\operatorname{T}}DP^{\operatorname{T}}\widehat{P})\Big{]}$
$\displaystyle=\operatorname{tr}(\widetilde{P}^{\operatorname{T}}D)-\Big{[}\|\widehat{P}^{\operatorname{T}}D\|_{\operatorname{tr}}-\operatorname{tr}(\widehat{P}^{\operatorname{T}}DP^{\operatorname{T}}\widehat{P})\Big{]},$

where we have used
$\|\widehat{P}^{\operatorname{T}}D\|_{\operatorname{tr}}=\operatorname{tr}(\widetilde{P}^{\operatorname{T}}D)$.
Note that
$\|\widehat{P}^{\operatorname{T}}D\|_{\operatorname{tr}}-\operatorname{tr}(\widehat{P}^{\operatorname{T}}DP^{\operatorname{T}}\widehat{P})\geq
0$ because

$\operatorname{tr}(\widehat{P}^{\operatorname{T}}DP^{\operatorname{T}}\widehat{P})\leq\|\widehat{P}^{\operatorname{T}}DP^{\operatorname{T}}\widehat{P}\|_{\operatorname{tr}}\leq\|\widehat{P}^{\operatorname{T}}D\|_{\operatorname{tr}}\|P^{\operatorname{T}}\widehat{P}\|_{2}\leq\|\widehat{P}^{\operatorname{T}}D\|_{\operatorname{tr}}.$

Finally use $0<s_{k}(B)\leq b\leq S_{k}(B)$ and $0<s_{k}(B)\leq\hat{b}\leq
S_{k}(B)$ to claim (F.6) from (F.5). ∎

### References

  * [1] P.-A. Absil, R. Mahony, and R. Sepulchre.  Optimization Algorithms On Matrix Manifolds.  Princeton University Press, Princeton, NJ, 2008.
  * [2] E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen.  LAPACK Users’ Guide.  SIAM, Philadelphia, 3rd edition, 1999.
  * [3] Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst (editors).  Templates for the solution of Algebraic Eigenvalue Problems: A Practical Guide.  SIAM, Philadelphia, 2000.
  * [4] Z. Bai, R.-C. Li, and D. Lu.  Sharp estimation of convergence rate for self-consistent field iteration to solve eigenvector-dependent nonlinear eigenvalue problems.  SIAM J. Matrix Anal. Appl., 43(1):301–327, 2022.
  * [5] Z. Bai and D. Lu.  Variational characterization of monotone nonlinear eigenvector problems and geometry of self-consistent-field iteration.  SIAM J. Matrix Anal. Appl., 46(1):84–111, 2024.
  * [6] J. Balogh, T. Csendes, and T. Rapcs’a.  Some global optimization problems on Stiefel manifolds.  J. Global Optim., 30:91–101, 2004.
  * [7] P. Benner and X. Liang.  Convergence analysis of vector extended locally optimal block preconditioned extended conjugate gradient method for computing extreme eigenvalues.  Numer. Linear Algebra Appl., 29(6):e2445, 2022.  24 pages.
  * [8] R. Bhatia.  Matrix Analysis.  Graduate Texts in Mathematics, vol. 169. Springer, New York, 1996.
  * [9] P. Birtea, I. Caşu, and D. Comănescu.  First order optimality conditions and steepest descent algorithm on orthogonal Stiefel manifolds.  Opt. Lett., 13:1773–1791, 2019.
  * [10] M. Bolla, G. Michaletzky, G. Tusnády, and M. Ziermann.  Extrema of sums of heterogeneous quadratic forms.  Linear Algebra Appl., 269(1):331–365, 1998.
  * [11] I. Borg and J. Lingoes.  Multidimensional Similarity Structure Analysis.  Springer-Verlag, New York, 1987.
  * [12] N. Boumal, B. Mishra, P.-A. Absil, and R. Sepulchre.  Manopt, a Matlab toolbox for optimization on manifolds.  J. Mach. Learning Res., 15(42):1455–1459, 2014.
  * [13] S. Boyd and L. Vandenberghe.  Convex Optimization.  Cambridge University Press, Cambridge, UK, 2004.
  * [14] Y. Cai, L.-H. Zhang, Z. Bai, and R.-C. Li.  On an eigenvector-dependent nonlinear eigenvalue problem.  SIAM J. Matrix Anal. Appl., 39(3):1360–1382, 2018.
  * [15] M. T. Chu and N. T. Trendafilov.  The orthogonally constrained regression revisited.  J. Comput. Graph. Stat., 10(4):746–771, 2001.
  * [16] J. P. Cunningham and Z. Ghahramani.  Linear dimensionality reduction: Survey, insights, and generalizations.  J. Mach. Learning Res., 16:2859–2900, 2015.
  * [17] J. Demmel.  Applied Numerical Linear Algebra.  SIAM, Philadelphia, PA, 1997.
  * [18] A. Edelman, T. A. Arias, and S. T. Smith.  The geometry of algorithms with orthogonality constraints.  SIAM J. Matrix Anal. Appl., 20(2):303–353, 1999.
  * [19] L. Eldén and H. Park.  A procrustes problem on the Stiefel manifold.  Numer. Math., 82:599–619, 1999.
  * [20] K. Fan.  On a theorem of Weyl concerning eigenvalues of linear transformations. I.  Proc. Natl. Acad. Sci. USA, 35(11):pp. 652–655, 1949.
  * [21] R. A. Fisher.  The use of multiple measurements in taxonomic problems.  Ann. Eugenics, 7(2):179–188, 1936.
  * [22] B. Gao, X. Liu, X. Chen, and Y.-X. Yuan.  A new first-order algorithmic framework for optimization problems with orthogonality constraints.  SIAM J. Optim., 28(1):302–332, 2018.
  * [23] G. Golub and Q. Ye.  An inverse free preconditioned Krylov subspace methods for symmetric eigenvalue problems.  SIAM J. Sci. Comput., 24:312–334, 2002.
  * [24] G. H. Golub and C. F. Van Loan.  Matrix Computations.  Johns Hopkins University Press, Baltimore, Maryland, 4th edition, 2013\.
  * [25] J. C. Gower and G. B. Dijksterhuis.  Procrustes Problems.  Oxford University Press, New York, 2004.
  * [26] N. J. Higham.  Functions of Matrices: Theory and Computation.  Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2008.
  * [27] P. Hohenberg and W. Kohn.  Inhomogeneous electron gas.  Phys. Rev., 136:B864–B871, 1964.
  * [28] R. A. Horn and C. R. Johnson.  Topics in Matrix Analysis.  Cambridge University Press, Cambridge, 1991.
  * [29] R. A. Horn and C. R. Johnson.  Matrix Analysis.  Cambridge University Press, New York, NY, 2nd edition, 2013.
  * [30] J. R. Hurley and R. B. Cattell.  The Procrustes program: producing direct rotation to test a hypothesized factor structure.  Comput. Behav. Sci., 7:258–262, 1962.
  * [31] A. Imakura, R.-C. Li, and S.-L. Zhang.  Locally optimal and heavy ball GMRES methods.  Japan J. Indust. Appl. Math., 33:471–499, 2016.
  * [32] C. Kanzow and H.-D. Qi.  A QP-free constrained Newton-type method for variational inequality problems.  Math. Program., 85:81–106, 1999.
  * [33] A. V. Knyazev.  Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method.  SIAM J. Sci. Comput., 23(2):517–541, 2001.
  * [34] W. Kohn and L. J. Sham.  Self-consistent equations including exchange and correlation effects.  Phys. Rev., 140:A1133–A1138, 1965.
  * [35] J. Kovač-Striko and K. Veselić.  Some remarks on the spectra of Hermitian matrices.  Linear Algebra Appl., 145:221–229, 1991.
  * [36] R. Lehoucq, D. C. Sorensen, and C. Yang.  ARPACK User’s Guide.  SIAM, Philadelphia, 1998.
  * [37] R.-C. Li.  A perturbation bound for the generalized polar decomposition.  BIT, 33:304–308, 1993.
  * [38] R.-C. Li.  New perturbation bounds for the unitary polar factor.  SIAM J. Matrix Anal. Appl., 16:327–332, 1995.
  * [39] R.-C. Li.  Accuracy of computed eigenvectors via optimizing a Rayleigh quotient.  BIT, 44(3):585–593, 2004.
  * [40] R.-C. Li.  Matrix perturbation theory.  In L. Hogben, R. Brualdi, and G. W. Stewart, editors, Handbook of Linear Algebra, page Chapter 21. CRC Press, Boca Raton, FL, 2nd edition, 2014\.
  * [41] R.-C. Li.  Rayleigh quotient based optimization methods for eigenvalue problems.  In Z. Bai, Weiguo Gao, and Yangfeng Su, editors, Matrix Functions and Matrix Equations, volume 19 of Series in Contemporary Applied Mathematics, pages 76–108. World Scientific, Singapore, 2015.
  * [42] X. Liang and R.-C. Li.  On generalizing trace minimization principles, II.  Linear Algebra Appl., 687:8–37, 2024.
  * [43] X. Liang, R.-C. Li, and Z. Bai.  Trace minimization principles for positive semi-definite pencils.  Linear Algebra Appl., 438:3085–3106, 2013.
  * [44] X. Liang, L. Wang, L.-H. Zhang, and R.-C. Li.  On generalizing trace minimization principles.  Linear Algebra Appl., 656:483–509, 2023.
  * [45] X.-G. Liu, X.-F. Wang, and W.-G. Wang.  Maximization of matrix trace function of product Stiefel manifolds.  SIAM J. Matrix Anal. Appl., 36(4):1489–1506, 2015.
  * [46] D. Lu and R.-C. Li.  Locally unitarily invariantizable NEPv and convergence analysis of SCF.  Math. Comp., 2024.  Published electronically: January 9, 2024.
  * [47] J. Moré and D. Sorensen.  Computing a trust region step.  SIAM J. Sci. Statist. Comput., 4(3):553–572, 1983.
  * [48] T. Ngo, M. Bellalij, and Y. Saad.  The trace ratio optimization problem for dimensionality reduction.  SIAM J. Matrix Anal. Appl., 31(5):2950–2971, 2010.
  * [49] F. Nie, R. Zhang, and X. Li.  A generalized power iteration method for solving quadratic problem on the Stiefel manifold.  SCIENCE CHINA Info. Sci., 60(11):1–10, 2017.
  * [50] J. Nocedal and S. Wright.  Numerical Optimization.  Springer, 2nd edition, 2006.
  * [51] B. N. Parlett.  The Symmetric Eigenvalue Problem.  SIAM, Philadelphia, 1998.  This SIAM edition is an unabridged, corrected reproduction of the work first published by Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1980\.
  * [52] B. T. Polyak.  Introduction to Optimization.  Optimization Software, New York, 1987.
  * [53] P. Quillen and Q. Ye.  A block inverse-free preconditioned Krylov subspace method for symmetric generalized eigenvalue problems.  J. Comput. Appl. Math., 233(5):1298–1313, 2010.
  * [54] T. Rapcsák.  On minimization on Stiefel manifolds.  European J. Oper. Res., 143(2):365–376, 2002.
  * [55] J. D. Rutter.  A serial implementation of Cuppen’s divide and conquer algorithm for the symmetric eigenvalue problem.  Technical Report UCB/CSD-94-799, EECS Department, University of California, Berkeley, February 1994.
  * [56] Y. Saad.  Numerical Methods for Large Eigenvalue Problems.  Manchester University Press, Manchester, UK, 1992.
  * [57] Y. Saad, J. R. Chelikowsky, and S. M. Shontz.  Numerical methods for electronic structure calculations of materials.  SIAM Rev., 52(1):3–54, 2010.
  * [58] G. W. Stewart and J. G. Sun.  Matrix Perturbation Theory.  Academic Press, Boston, 1990.
  * [59] J. G. Sun.  Matrix Perturbation Analysis.  Graduate Texts (Academia, Sinica). Science Publisher, Beijing, 2nd edition, November 2001.  in Chinese.
  * [60] I. Takahashi.  A note on the conjugate gradient method.  Inform. Process. Japan, 5:45–49, 1965.
  * [61] J. M. F. Ten Berge.  Generalized approaches to the MAXBET problem and the MAXDIFF problem, with applications to canonical correlations.  Psychometrika, 53(4):487–494, 1984.
  * [62] Z. Teng and R.-C. Li.  Variations of orthonormal basis matrices of subspaces.  Numer. Alg., Contr. Optim., 2024.  to appear.
  * [63] J. P. Van de Geer.  Linear relations among $k$ sets of variables.  Psychometrika, 49(1):70–94, 1984.
  * [64] J. von Neumann.  Some matrix-inequalities and metrization of matrix-space.  Tomck. Univ. Rev., 1:286–300, 1937.
  * [65] L. Wang, B. Gao, and X. Liu.  Multipliers correction methods for optimization problems over the Stiefel manifold.  CSIAM Trans. Appl. Math., 2(3):508–531, 2021.
  * [66] L. Wang, L.-H. Zhang, and R.-C. Li.  Maximizing sum of coupled traces with applications.  Numer. Math., 152:587–629, 2022.  doi.org/10.1007/s00211-022-01322-y.
  * [67] L. Wang, L.-H. Zhang, and R.-C. Li.  Trace ratio optimization with an application to multi-view learning.  Math. Program., 201:97–131, 2023.  doi.org/10.1007/s10107-022-01900-w.
  * [68] H. F. Weinberger.  Variational Methods for Eigenvalue Approximation, volume 15 of CBMS-NSF Regional Conference Series in Applied Mathematics.  SIAM, Philadelphia, 1974.
  * [69] Z. Wen and W. Yin.  A feasible method for optimization with orthogonality constraints.  Math. Program., 142(1-2):397–434, 2013.
  * [70] C. Yang, J. C. Meza, B. Lee, and L.-W. Wang.  KSSOLV—a MATLAB toolbox for solving the Kohn-Sham equations.  ACM Trans. Math. Software, 36(2):1–35, 2009.
  * [71] M. Yang and R.-C. Li.  Heavy ball flexible GMRES method for nonsymmetric linear systems.  J. Comp. Math., 40(5):715–731, 2021.
  * [72] L.-H. Zhang, L.-Z. Liao, and M. K. Ng.  Fast algorithms for the generalized Foley-Sammon discriminant analysis.  SIAM J. Matrix Anal. Appl., 31(4):1584–1605, 2010.
  * [73] L.-H. Zhang, L.-Z. Liao, and M. K. Ng.  Superlinear convergence of a general algorithm for the generalized Foley-Sammon discriminant analysis.  J. Optim. Theory Appl., 157(3):853–865, 2013.
  * [74] L.-H. Zhang, W. H. Yang, C. Shen, and J. Ying.  An eigenvalue-based method for the unbalanced Procrustes problem.  SIAM J. Matrix Anal. Appl., 41(3):957–983, 2020.
  * [75] L.-H. Zhang and R.-C. Li.  Maximization of the sum of the trace ratio on the Stiefel manifold, I: Theory.  SCIENCE CHINA Math., 57(12):2495–2508, 2014.
  * [76] L.-H. Zhang and R.-C. Li.  Maximization of the sum of the trace ratio on the Stiefel manifold, II: Computation.  SCIENCE CHINA Math., 58(7):1549–1566, 2015.
  * [77] L.-H. Zhang, L. Wang, Z. Bai, and R.-C. Li.  A self-consistent-field iteration for orthogonal canonical correlation analysis.  IEEE Trans. Pattern Anal. Mach. Intell., 44(2):890–904, 2022.
  * [78] Z. Zhang and K. Du.  Successive projection method for solving the unbalanced Procrustes problem.  SCIENCE CHINA Math., 49(7):971–986, 2006.
  * [79] Z. Zhang, Z. Zhai, and L. Li.  Uniform projection for multi-view learning.  IEEE Trans. Pattern Anal. Mach. Intell., 39(8):1675–1689, 2017\.
  * [80] H. Zhao, Z. Wang, and F. Nie.  Orthogonal least squares regression for feature extraction.  Neurocomputing, 216:200–207, 2016.
  * [81] Y. Zhou and R.-C. Li.  Bounding the spectrum of large Hermitian matrices.  Linear Algebra Appl., 435:480–493, 2011.