January 2023

Should we worry about renormalons in the $\epsilon$-expansion?

E. Brézina)

a) Laboratoire de Physique de l’Ecole normale supérieure, ENS, Université PSL,
CNRS, Sorbonne Université, Université de Paris, F-75005 Paris, e-mail:
<EMAIL_ADDRESS>

Submitted for publication in a book commemorating
Michael Fisher
(Amnon Aharony, Ora Entin, David Huse, Leo Radzihovsky, editors)

Abstract Turning the divergent $\epsilon$-expansion into a numerically
sensible algorithm, relies on the knowledge of the behaviour of the large
order contributions. Two different pictures are known to compete there. The
first one was based on Lipatov’s instantons, which is known to deal with the
multiplicity of Feynman diagrams which grows factorially at high orders.
However this was challenged by ’t Hooft’s renormalons who pointed out that
renormalization could yield a similar growth through one single diagram. We
study here a well-known model, the $O(N)$ model, in the large $N$ limit. The
reason for returning to this familiar model, is that it deals with diagrams
known to give renormalon effects.Through an explicit analytic result, we find
no sign of a non-analyticity of perturbation theory due to these renormalons.

##  1 Introduction

A little more than fifty years ago K.Wilson and M.Fisher introduced the
$\epsilon$-expansion in the celebrated article [1] Critical Exponents in 3.99
Dimensions. This article has had a considerable influence both in the area of
critical phenomena and in quantum field theory in general. However in
practice, if the first two terms of the expansion provided often a reasonable
approximation to the measured indices in dimension three, the situation
deteriorated if one pushed the procedure to higher orders [2]. Clearly the
expansion looked divergent, it is believed to be at best asymptotic, and in
fact limited in its applicability. Of course summation procedure, such as
Borel transforms, Padé approximants, could be tried, but in the absence of
further indication, it was a blind shot. Therefore at the time analytic
calculations, and computer simulations, appeared to be limited in their
ability of reproducing precision measurements by the renormalization group
approach.

The situation changed significantly in 1977 with Lipatov approach to the
characterization of large orders in pertubation theory [3]. His instanton
method , which he developed for the field theory $g\phi^{4}$ in dimension
four, predicted a large order behaviour of the perturbation expansion of the
various correlation functions, $\beta$-function, etc, of the form
$g^{k}k!(-a)^{k}k^{b}c$ for large $k$, with calculable coefficients $a,b,c$ .
I shall recall below why this information, the explicit knowledge of $a,b,c$,
is essential to extract a numerically sensible result from such a divergent
series. With Le Guillou and Zinn-Justin [4] we first checked these results in
lower dimensions. For instance the one-dimensional quartic anharmonic
oscillator had been extensively studied by Bender and Wu [5] who had found, on
the basis of a WKB method, that indeed perturbation theory diverged with a
$k!$ growth as above. We verified that the instanton method reproduced exactly
what they had found. Extension to field theory in higher dimensions was
similar [4] except that the instanton solution is not known in analytic form
except in dimension four [6]. In dimensions lower than four the interpretation
of these results is simple : at a given order all Feynman diagrams have the
same sign (we are considering an Euclidean field theory) and their number
grows proportionaly to $k!$. The situation is less transparent in dimension
four since diagrams require counterterms and subtractions. This is the central
point of the investigation that we try to carry in this paper.

The knowledge of the large order behaviour of the perturbative expansion was
the source of considerable improvements in the ability to get a sensible
answer from those rapidly divergent series [8]. In practice it led to
numerically convergent algorithms developed around Zinn-Justin and
collaborators[8, 9], Borel transform of the series, followed by conformal
mappings relying on the explicit knowledge of the coefficients $a.b,c$ here
above, were among the techniques that they used and examples, such as the
anharmonic oscillator, revealed that the divergent perturbation series could
lead, after those mappings, to many digits exact results.

In our 1977 paper [4] we tried to carry the large order knowledge of
perturbative series to the $\epsilon$-expansion. In most procedures one has to
perform a double expansion in $g$ and $\epsilon$. Since the fixed point
$g^{*}$ is of order $\epsilon$ one has to consider at order $k$, terms of
order $g^{k},g^{k-1}\epsilon,g^{k-2}\epsilon^{2},\cdots$ . Fortunately the
well-known technique of minimal subtraction allows one to avoid this double-
expansion. In this renormalization scheme [12] by definition the counter-terms
contain only poles in $\epsilon$ without any finite part. As a result one
obtains the renormalization group $\beta$ and $\gamma$ functions in dimension
$d$ from their four-dimensional counterpart

$\displaystyle\beta_{d}(g)=-\epsilon g+\beta_{4}(g)$
$\displaystyle\gamma_{d}(g)=\gamma_{4}(g)$ (1)

The fixed point $g^{*}$ is then $\epsilon$-expanded from

$\epsilon g^{*}=\beta_{4}(g^{*})$ (2)

($\beta_{4}(g)$ is a series starting at order two), and the critical exponent
$\eta$, which is (twice) the conformal anomaly of the $\phi$-field is then
expanded in $\epsilon$ from

$\eta=\gamma_{4}(g^{*})$ (3)

Therefore computing the $\epsilon$-expansion is reduced to four-dimensional
calculations. Using the 4D large-order calculation from Lipatov method we
obtained an estimate of the asymptotic orders in the $\epsilon$-expansion [4]
which was again of the $k!$-type as before. Then Zinn-Justin and co-workers
based a summation procedure on this large order behaviour [8, 9], and the
process looked nicely convergent : at least adding one more order in the
$\epsilon$-expansion improved the previous result, instead of destroying it as
the straight expansion does.

The confidence in this process was severely affected after ’t Hooft’s
discovery of a phenomenon [10], now called renormalons, which appears only in
renormalizable theories such as $\phi^{4}$ in dimension four (and not in lower
dimensions). His argument was based on the fact that one single diagram at
order $k$ could be proportional to $k!$ , whereas the $k!$ , for $d<4$,
resulted from their multiplicity. In a renormalizable theory a diagram, such
as the (renormalized) bubble diagram, grows logarithmically for a large
external momentum. A repeated insertion of such diagrams leads to an integral
with a log at k-th power, which gives after integration a $k!$. Clearly this
shed doubt on Lipatov’s estimate for large order, although ’t Hooft’s argument
didn’t show explicitely that it was wrong. It was not really clear either how
those renormalons affected quantitatively the actual calculation.

I have remained puzzled by this problem since then, wondering whether the
ambitious RG work based on the $\epsilon$-expansion, was simply an
approximation which should not be pushed too far. I want to present here a
simple well-known problem, namely the $O(N)$-model in the large-N limit. The
reason for returning to this familiar model is that it is clearly a candidate
for showing up renormalons effect, whereas this time we do not need an
instanton asymptotics since there are only a few diagrams of a given order. I
do not pretend that it solves the instanton-renormalon competition, but I
wonder on the basis of this calculation whereas renormalons affect the
perturbation series as badly as one could think.

##  2 The $O(N)$ model in dimension four

This, half a century-old, model consists of an $N$-component order parameter
$\phi_{a},a=1,\cdots,N$, with an interaction invariant under $O(N)$ [7]. This
model has been studied by hundreds of authors and elaborate techniques have
allowed to compute several orders in a $1/N$ expansion. We will limit
ourselves here to the leading large $N$ terms for the RG functions $\beta(g)$
and $\gamma(g)$ in the minimal subtraction scheme which allows to make easy
contact with the $\epsilon$-expansion. The reason for our interest is that
$\gamma(g)$ is of order $1/N$ and the expansion in powers of $g$ of this
leading term involves exactly the diagrams which have been identified as
generating renormalon singularities in the perturbative expansion. With the
help of previous results on critical indices we will try to understand what
the renormalons do to this expansion.

The model is given by an (Euclidean) action

$S=\int
d^{4}x[\frac{1}{2}\nabla\phi_{a}\nabla\phi_{a}+\frac{1}{2}m_{o}^{2}\phi^{2}+\frac{g_{0}}{4!N}(\phi^{2})^{2}]$
(4)

which one could regularize by an ultra-violet cut-off. For the reasons
mentioned above we prefer here the dimensional regularization by going to
$4-\epsilon$ dimensions, and then renormalize minimally by a coupling constant
renormalization $Z_{1}$ and a field rescaling $\phi=\sqrt{Z}\varphi$. We work
in the massless theory (critical temperature) and the action in terms of the
renormalized field reads (omitting mass counterterms)

$S=\int
d^{d}x[\frac{Z}{2}\nabla\varphi_{a}\nabla\varphi_{a}+\mu^{\epsilon}\frac{gZ_{1}}{4!N}(\varphi^{2})^{2}]$
(5)

Varying $\mu$ at fixed bare theory we obtain the standard renormalization
group functions of the Callan-Symanzik equation [11]

$\epsilon g+\beta(g)(1+g\frac{d}{dg}\ln(Z_{1}/Z^{2}))=0$ (6)
$\gamma(g)=\beta(g)\frac{d}{dg}\ln(Z)$ (7)

We are interested in computing the leading terms in a $1/N$ expansion of
$Z_{1}$ which is $O(N^{0})$ and $Z$ since $Z=1+O(1/N)$.

###  2.1 The $\beta$-function at order $N^{0}$

The coupling constant involving an explicit factor $1/N$, the leading terms
maximize diagrams with internal index-loops which provide a compensating
factor $N$. For the two-point function the leading diagrams are of order $1/N$
and the field renormalization $Z=1+O(1/N)$. For the four-point function the
leading diagrams consist of a string of ”bubbles” all of order $1/N$ and this
yields a vertex renormalization $Z_{1}$ of order $N^{0}$ which we now compute.

Let us begin with a bubble with external momentum $p$

$B(p)=\int\frac{d^{d}q}{q^{2}(p-q)^{2}}$ (8)

where we use the convention of omitting the usual geometric factor
$\frac{2\pi^{d/2}}{(2\pi)^{d}\Gamma(d/2)}$ included into a rescaling of $g$
that will be implied ; we take the scale factor $\mu$ as unit of momentum.

$q$$p-q$$B(p)=$

A standard calculation (using a Feynman parameter) yields

$\displaystyle B(p)=a(\epsilon)\frac{p^{-\epsilon}}{\epsilon},$ $\displaystyle
a(\epsilon)=\frac{1-\epsilon/2}{1-\epsilon}\
\frac{\Gamma^{3}(1-\epsilon/2)\Gamma(1+\epsilon/2)}{\Gamma(1-\epsilon)}=1+\frac{\epsilon}{2}+O(\epsilon^{2})$
(9)

i.e.

$B(p)=\frac{1}{\epsilon}+1/2-\ln{p}$ (10)

Then the four-point function, for external indices a,a,b,b is given by a
geometric series

$..$$\cdots$$...$$...$$+$$+$$...$$...$$...$$+\cdots$$\Gamma^{(4)}=$

$\displaystyle-\frac{1}{2}\Gamma^{(4)}=\frac{1}{N6}gZ_{1}-\frac{g^{2}Z_{1}^{2}}{N6^{2}}B(p)+\frac{g^{3}Z_{1}^{3}}{N6^{3}}B^{2}(p)+\cdots$
$\displaystyle=\frac{1}{N}\frac{gZ_{1}/6}{1+gZ_{1}/6\
B(p)}=\frac{1}{N}\frac{g/6}{1/Z_{1}+g/6\ B(p)}+O(1/N^{2})$ (11)

Note that the Z factor omitted here would give only a $1/N^{2}$ contribution.
Then taking

$1/Z_{1}=1-\frac{g}{6\epsilon}$ (12)

which satisfies the minimal subtraction rule, we obtain a finite
$\Gamma^{(4)}$ in the limit $\epsilon\to 0$, at order zero in $1/N$ . The
$\beta$-function follows immediately

$0=\epsilon g+\beta(g)[1+\frac{g}{6\epsilon-g}]$

i.e.

$\beta(g)=-\epsilon g+\frac{1}{6}g^{2}+O(1/N)$ (13)

One can check this result which is valid to all orders in $g$ , but zeroth
order in $1/N$, with the literature. We copy from Zinn-Justin’s book [13] in
which he used the minimal subtraction scheme

$\beta=-\epsilon g+\frac{N+8}{6N}g^{2}-\frac{3N+14}{12N^{2}}g^{3}+\cdots$ (14)

and higher terms in $g$ are of order $1/N$, which agrees with (13) when $N$
goes to infinity.

###  2.2 The two-point function

We are now considering the diagrams for the inverse two-point function
$\Gamma^{(2)}(p)$.

$p$$...$$p-q$$...$$\Gamma^{(2)}(p)=Zp^{2}-\ $

At order $g^{2}$ we have one inserted bubble diagram, thus

$\Gamma^{(2)}(p)=Zp^{2}-2\frac{(gZ_{1})^{2}}{N6^{2}}\int
d^{d}q\frac{B(q)}{(p-q)^{2}}+O(g^{3},1/N^{2})$ (15)

The integral over $q$ diverges as $-\frac{1}{8\epsilon}$, we will compute
explicitly the integral with a string of bubbles of arbitrary length
herefafter. So at order $g^{2}$

$Z=1-\frac{1}{144N}\frac{g^{2}}{\epsilon}+O(g^{3},1/N^{2})$ (16)

giving

$\gamma(g)=\frac{g^{2}}{72N}+O(g^{3},1/N^{2})$ (17)

.

Tot all orders in $g$ the string of bubbles gives

$\Gamma^{(2)}(p)=Zp^{2}-\frac{2}{N}\sum_{k=1}(\frac{-gZ_{1}}{6})^{k+1}\int
d^{d}q\frac{B^{k}(q)}{(p-q)^{2}}+O(1/N^{2})$ (18)

We will sum the series later, but it is interesting to study the finite order
$k$ dealing thus with the integral

$I_{k}=\int
d^{d}q\frac{B^{k}(q)}{(p-q)^{2}}=\frac{a^{k}(\epsilon)}{\epsilon^{k}}\int
d^{d}q\frac{q^{-k\epsilon}}{(p-q)^{2}}$ (19)

with $B(q)$ and $a(\epsilon)$ given in (2.1).

$I_{k}=$$p$$p-q$$\cdots\cdots$.

It is interesting to compute $I_{k}$ explicitly to understand what happens at
higher $k$’s. Standard techniques give

$\displaystyle I_{k}=$
$\displaystyle-\frac{k}{4(k+1)}\frac{a^{k}p^{2-\epsilon(k+1)}}{\epsilon^{k}}\frac{\Gamma^{2}(1-\epsilon/2)\Gamma(1-\epsilon(k+1)/2)\Gamma(1+\epsilon(k+2)/2)}{\Gamma(1-\epsilon(k+2)/2)\Gamma(1+\epsilon
k/2)}$ (20)
$\displaystyle\times\frac{(1-\epsilon/2)}{(1-\epsilon(k+2)/4)(1-\epsilon(k+2)/2)}$
$\displaystyle=-\frac{k}{4(k+1)}\frac{p^{2-\epsilon(k+1)}}{\epsilon^{k}}[1+(5k/4+1)\epsilon+O(\epsilon^{2})]$

The calculations from thereon are straightforward, we just have to expand in
powers of $\epsilon$ the various explicit functions which appear in (19,2.1)
and chose $Z$ to cancel all the poles in $\epsilon$. For instance at order
$g^{3}$ if we take

$Z=1-\frac{g^{2}}{N144\epsilon}-\frac{g^{3}}{N6^{4}\epsilon^{2}}(1-\epsilon/4)+O(g^{4}/N,1/N^{2})$
(21)

we verify that $\Gamma^{(2)}$ is finite up to this order, as renormalization
theory implies,

$\Gamma^{(2)}(p)=p^{2}-\frac{g^{2}}{144N}p^{2}(2\ln
p-7/8)-\frac{9g^{3}}{6^{4}N}p^{2}\ln^{2}p+O(g^{4}/N).$ (22)

From (21) we obtain immediately at leading order $1/N$

$\gamma(g)=\frac{1}{72N}[g^{2}-\frac{1}{24}g^{3}+O(g^{4})]$ (23)

which agrees for $N$ large with the result in [13]

$\gamma(g)=\frac{N+2}{72N^{2}}g^{2}[1-\frac{N+8}{24N}g+\frac{5(-N^{2}+18N+100)}{576N^{2}}g^{2}]+\cdots$
(24)

We can proceed in this fashion to all orders in $g$, but it is tedious. One
can also sum the series (18) but it is not simple either to extract $Z$ from
the sum.

$\Gamma^{(2)}=Zp^{2}-\frac{g^{2}Z_{1}}{18N}\int\frac{d^{d}q}{(p-q)^{2}}\frac{1}{\frac{1}{Z_{1}B(q)}+g/6}$
(25)

Fortunately previous results on the $1/N$ expansion of this model allow us to
recover $\gamma(g)$, at order $1/N$, to all orders in $g$ as explained in the
next section.

##  3 Where are the renormalons?

If instead of dimensional regularization we had stayed in four dimensions with
an ultra-violet cut-off $\Lambda$ we could have computed the same diagrams The
bubble-diagram (8),

$\int_{\Lambda}\frac{d^{4}v}{(v-q)^{2}v^{2}}$ (26)

behaves as $\ln{q/\Lambda}$ at small momentum. Inserted in the two-point the
$k$ -th iterated bubble behaves as $\ln^{k}{q/\Lambda}$ and inserted in the
two-point function it yields the integral

$\int_{\Lambda}d^{4}q[\frac{1}{(p-q)^{2}}-\frac{1}{q^{2}}]\ln^{k}{q/\Lambda}$
(27)

where we have explicited the zero-momentum subtraction of the massless theory
(which automatically vanished in the minimal scheme). The resulting integral
is porportional to $p^{2}$ and it yields an integral over $q$ which is infra-
red singular in the $p$ small region. Taking $\ln q/\Lambda=-x$ the singular
part is given by a power of a logarithmic singularity in $p$ , with a
coefficient which behaves for large $x$ as $\int dxe^{-x}(-x)^{k}$, i.e. a
factorial growth with alternating signs. This is the argument for a
perturbation expansion exhibiting (infra-red) renormalons [7].

Our goal is to compute the renormalized correlation functions, the scaling
limit of the theory in which distance are much larger than the lattice spacing
$\Lambda^{-1}$. We have seen in the previous section how complex is the
interplay between the diagrams and the counter-terms and this is not
transparent in the above cut-off regularized theory. So let us return to the
results of the previous section within dimensional regularization. Looking
back at the $k$-th order, i.e. the integral (20), we see explicitly that the
$k$ large and $\epsilon$ small limits do not commute. The renormalization
procedure is strictly defined as $\epsilon$ goes to zero first, cancelling the
poles in $\epsilon$ through the poles coming from $Z$ and $Z_{1}$. It is only
after we have removed those singular terms that we may examine the asymptotic
behavior for large $k$. In the minimal subtraction scheme that we have
followed here, this is done by cancelling all the poles and multiple poles in
$\epsilon$ occuring in the results such as (20) with the poles in $Z$ and
$Z_{1}$.

Could that produce the same renormalon-$k!$ ? Indeed if we return to (20) the
factor $\frac{1}{\epsilon^{k}}p^{2-\epsilon(k+1)}$ will end up expanded $k$
times in powers of $\epsilon$ by the time all the subtractions manage to
produce a finite four-dimensional theory (i.e, $\epsilon=0$). This might yield
a behaviour of the two-point function at order $k$ with a term
$p^{2}[-(k+1)\ln p]^{k}$ and of course $(-k)^{k}\simeq(-a)^{k}k!$, but this is
far from obvious, the algebra could produce a $1/k!$ which would kill this
would-be renormalon. However our goal here is to understand the large orders
of the $\epsilon$-expansion and that relies on the expansion of the
renormalization group $\gamma$ function. There we will see that there is no
room for a renormalon large order behavior.

##  4 An explicit solution through earlier results

Since the removal of the poles in $\epsilon$ is increasingly more cumbersome
when the order increases, fortunately we can call on previous results on the
large $N$ limit to bypass this long algebra. In fact there are better ways of
dealing with the $1/N$ expansion, like adding to the action a Lagrange
multiplier $\lambda({\varphi}^{2}-\psi)$, replacing the quartic term in
$\varphi$ by $\psi^{2}$ and tracing out the Gaussian $\varphi$’s. The
expansion around the saddle-point of the resulting $(\lambda,\psi)$ action
yields the $1/N$-expansion [13, 7]. The reason for not following this
procedure here is that we needed to stick to the minimal subtraction scheme.
Several terms of the $1/N$ expansion of the critical exponents have been
computed for arbitrary dimensions, much more than what we needed here. In
particular we find in [7] the critical exponent $\eta$ at leading order, and
copy here the result

$\eta=\frac{1}{N}[\frac{\epsilon^{2}}{2}\frac{\Gamma(1-\epsilon)}{\Gamma^{3}(1-\epsilon/2)\Gamma(1+\epsilon/2)}\frac{1-\epsilon}{(1-\epsilon/4)(1-\epsilon/2)}]+O(\frac{1}{N^{2}})$
(28)

But we know that $\eta=\gamma(g^{*})$ and $g^{*}=6\epsilon+O(1/N)$. Therefore
we obtain the RG function $\gamma(g)$ by replacing $\epsilon$ by $g/6$ in
$\eta$ providing the result to all orders in $g$

$\gamma(g)=\frac{g^{2}}{72N}[\frac{\Gamma(1-g/6)}{\Gamma^{3}(1-g/12)\Gamma(1+g/12)}\frac{1-g/6}{(1-g/24)(1-g/12)}]+O(\frac{1}{N^{2}})$
(29)

This is the result valid to all orders in $g$, first order in $1/N$, that we
were looking for. One verifies easily that this, expanded in $g$, reproduces
what we had found before at low order (23).

##  5 Concluding remarks

  * •

The result (29) is analytic in $g$ in the neighbourhood of the origin : it is
meromorphic in $g$ with the closest singularity at $g=12$. Had the large
orders of the expansion in $g$ be growing factorially that the result would
not be analytic at $g=0$. This is reminiscent of what is familiar in matrix
models : the matrix integral is not analytic in $g$, the coefficient in $g\rm
TrM^{4}$. However if one considers the large $N$ limit, and the successive
terms of the $1/N$-expansion, every term of the expansion is analytic at
$g=0$. It has been argued by previous authors that renormalons are not present
in $\varphi^{4}_{4}$ [16] : the explicit calculation performed here confirms
this position.

  * •

The potentially dangerous renormalons do not show up in the final result

  * •

We have not shown that other correlation functions, other than the one we have
computed here, could not show renormalons, but it seems likely, in view of
what we did, that they are simply absent at first order in $1/N$ and we are
inclined to believe that this remains true to all orders in a $1/N$ expansion.

  * •

We have not shown that renormalons would not show up at fixed $N$, but the
argument in their favour being a priori operative, but finally absent, for the
case that we have considered here, we see no reason to believe that they spoil
the old result [4] on the large orders of the $\epsilon$-expansion.

  * •

The model that we have considered, was considered as a candidate for infra-red
renormalons. We have nothing to say on potential UV renormalons as in gauge
theories.

  * •

Many contemporary scientists probably consider that the problem discussed here
is obsolete ; who needs the $\epsilon$-expansion given the magnificent
precision of the conformal bootstrap [14], which defeats earlier methods in
their accuracy at predicting critical exponents ? However I believe that there
are many problems of interest in the scaling region for which the tools of
conformal bootstrap are not (not yet?) available. For instance, the universal
scaling equation of state, still relies on expansions : it was my first
article (with Wallace and Wilson) [15] using the $\epsilon$-expansion, fifty
years ago!

Acknowledgement I thank Giorgio Parisi for a discussion which led me to
reconsider this ancient story.

## References

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  * [2] E. Brézin, JC. Le Guillou, J. Zinn-Justin, J. and BG. Nickel, Phys. Lett. 44A (1973) 227 Higher order contributions to critical exponents
This was only $O(\epsilon^{3})$ ; nowadays five or six terms of the
$\epsilon$-expansion have been computed, confirming its numerical divergence.

  * [3] LN. Lipatov, Sov. Phys. JETP 45, 216 (1977) Divergence of the perturbation-theory series and the quasi-classical theory
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  * [8] JC Le Guillou and J. Zinn-Justin, Phys. Rev. B 21, 3976 (1980) Critical exponents from field theory
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K. Symanzik, Comm. Math. Phys.18, 227 (1970), Small distance behaviour in
field theory and power counting.

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# On a heuristic point of view concerning the optical activity

Chun-Fang Li<EMAIL_ADDRESS>Department of Physics, Shanghai University, 99
Shangda Road, 200444 Shanghai, China  Zhi-Juan Hu<EMAIL_ADDRESS>Department of Physics, Shanghai Normal University, 100 Guilin Road, 200233
Shanghai, China

###### Abstract

Motivated by a recent finding that Fresnel’s phenomenological description of
the optical activity in the chiral medium is not self-consistent, we conduct a
thorough investigation into the nature of the polarization of a plane light
wave. We demonstrate that the polarization of light is the reflection of one
of its quantum-mechanical properties, called the quasi-spin. Unexpectedly, the
quasi-spin is not an observable with respect to the laboratory coordinate
system. Instead, it is with respect to the momentum-dependent local coordinate
system. The representative operators for the quasi-spin are the Pauli
matrices. The wavefunction is the Jones vector. In order to completely
determine a state of polarization, two different kinds of degrees of freedom
are needed. One is the degrees of freedom to characterize the state of quasi-
spin. They are the Stokes parameters, the expectation values of the Pauli
matrices in the state described by the Jones vector. The other is the degrees
of freedom to specify the local coordinate system, including the propagation
direction and an angle of rotation about it. Accordingly, there are two
independent mechanisms to change the state of polarization. One is to change
the state of quasi-spin in a fixed local coordinate system. This is the
traditional mechanism that can be expressed as an SU(2) rotation of the Jones
vector. The other is to change the local coordinate system with the state of
quasi-spin remaining fixed in it. At last we show that it is the newly-
identified mechanism that accounts for the optical activity.

##  I Introduction

Optical activity is one of the most fundamental physical phenomena. It refers
to the ability of a medium to rotate the orientation of the polarization
ellipse of an elliptically polarized light wave that passes through it. Such
an ability is associated with the chirality of the molecular or crystalline
configuration of the optically-active medium [1, 2]. A helically-coiled
optical fiber also shows optical activity [3, 4, 5, 6]. It has been widely
accepted [7, 8, 9, 10, 11, 3, 4, 5, 12, 13, 6] that optical activity lies with
the allogyric birefringence [14] or the circular birefringence [15]; that is,
the right-handed circularly polarized (RCP) and left-handed circularly
polarized (LCP) waves in the chiral medium propagate at different velocities.
Although the assumption of the existence of circular birefringence, first
suggested by Fresnel in 1825, succeeds in describing the rotation of the
polarization plane of a linearly polarized wave, it fails to account for the
propagation velocity of a linearly polarized wave. Recently we proved [16]
that circular birefringence does not exist at all in the chiral medium.

The fact is that there exist two orthonormal linearly polarized modes in the
chiral medium, just like in the achiral medium. They have the same propagation
velocity. One can take them as base modes to expand an elliptically polarized
wave. As a result, any elliptically polarized wave, including circularly
polarized waves, propagates at the velocity of the linearly polarized waves.
The key point is that the two orthonormal linearly polarized waves in the
chiral medium are rotated in the same way. Such a rotation is expressed by
their rotatory polarization vectors. In fact, it is their rotatory
polarization vectors that are orthogonal to each other. When they are taken as
base modes to expand an elliptically polarized wave, their polarization
vectors serve as the polarization bases of the elliptically polarized wave.
This means that the optical activity amounts to the rotation of the
polarization bases, without changing the expansion coefficients that
constitute the Jones vector. In other words, the optical activity implies a
new mechanism to change the polarization of light. It is improper to interpret
the optical activity as the change of the Jones vector [14, 17, 18]. The
purpose of the present paper is to reveal the physics behind this mechanism.
The contents are arranged as follows.

In order to have a clearer look at the problem that we are faced up with, we
briefly review in Section II the main results that we obtained in Ref. [16]
from a slightly different point of view. In particular, we show that what
completely describes the state of polarization of a plane wave is its
polarization vector rather than its Jones vector. In Section III we prove from
the point of view of measurement that two different kinds of degrees of
freedom are needed to determine a polarization vector. One is the Stokes
parameters that are solely determined by the Jones vector. The other is the
degrees of freedom to specify a local coordinate system in which the Stokes
parameters as well as the Jones vector are defined, including the propagation
direction and an angle of rotation about it. The Stokes parameters are shown
in Section IV to be the expectation values of the representative operators for
a quantum-mechanical observable, called the quasi-spin, in a state described
by the Jones vector. They constitute the so-called Poincaré vector [19] and
take the role of characterizing the state of quasi-spin in the local
coordinate system. On this basis, it is argued in Section V that the state of
polarization is the reflection of the state of quasi-spin in the local
coordinate system. It not only has the property of the quasi-spin state but
also conveys the information of the local coordinate system. Thus there are,
in principle, two independent mechanisms to change the state of polarization.
One is to change the state of quasi-spin in a fixed local coordinate system.
This is the traditional mechanism that can be interpreted in terms of the
change of the Jones vector. The other is to change the local coordinate system
with the state of quasi-spin remaining fixed in it. These two mechanisms are
further discussed in Section VI. We show that the traditional mechanism is
responsible for implementing the change of the polarization state by use of
waveplates. But it is the newly-identified mechanism that accounts for the
rotation of the polarization bases in the chiral medium. Section VII concludes
the paper.

##  II Brief review of new description of optical activity

Chiral objects provide a coupling of the magnetic field to the electric
polarizability and of the electric field to the magnetic polarizability. This
is described macroscopically by the constitutive relations. The well-accepted
constitutive relations for an isotropic and lossless chiral medium can be
written as follows [20, 21],

$\displaystyle\mathbf{D}$
$\displaystyle=\varepsilon\mathbf{E}-g\partial{\mathbf{H}}/\partial{t},$ (1a)
$\displaystyle\mathbf{B}$
$\displaystyle=\mu\mathbf{H}+g\partial{\mathbf{E}}/\partial{t},$ (1b)

where $\mathbf{E}$, $\mathbf{H}$, $\mathbf{D}$, and $\mathbf{B}$ are, as
usual, the vectors of electric field, magnetic field, electric displacement,
and magnetic induction, respectively, $\varepsilon$ is the permittivity, $\mu$
is the permeability, and the pseudo-scalar constant $g$ is the gyrotropic
parameter. Let us first prove that circular birefringence does not exist in
the chiral medium.

###  II.1 No circular birefringence

Traditionally, the RCP and LCP plane waves in the chiral medium were referred
to as the characteristic modes with different propagation velocities. Suppose
that the propagation direction is along the positive $z$-axis. According to
Refs. [14, 15], the electric fields of the RCP and LCP plane waves of
normalized amplitude and of the same frequency $\omega$ take the form of

$\displaystyle\mathbf{E}_{R}(z,t)$
$\displaystyle=\frac{1}{\sqrt{2}}(\bar{x}+i\bar{y})e^{i(k_{R}z-\omega t)},$
(2a) $\displaystyle\mathbf{E}_{L}(z,t)$
$\displaystyle=\frac{1}{\sqrt{2}}(\bar{x}-i\bar{y})e^{i(k_{L}z-\omega t)},$
(2b)

where $\bar{x}$ and $\bar{y}$ are the unit vectors along the $x$\- and
$y$-axes, respectively, and $k_{R}$ and $k_{L}$ are their “wave numbers”. When
constitutive relations (1) are taken into consideration, it can be shown [20,
21] that

$\begin{split}k_{R}&=k-\tau,\\\ k_{L}&=k+\tau,\end{split}$ (3)

where $k=(\varepsilon\mu)^{1/2}\omega$ and $\tau=-g\omega^{2}$. It is noted
that $\tau$ depends on the gyrotropic parameter $g$; whereas $k$ does not.
Eqs. (2a) and (2b) were commonly interpreted to express two waves that
propagate at velocities $\frac{\omega}{k_{R}}$ and $\frac{\omega}{k_{L}}$,
respectively. The problems with such an interpretation are as follows.

It is true that the coherent superposition of $\mathbf{E}_{R}$ and
$\mathbf{E}_{L}$ in (2) can give rise to linearly polarized waves. In
particular, the following two superpositions are linearly-polarized,

$\begin{split}\mathbf{E}_{1}&=\frac{\mathbf{E}_{R}+\mathbf{E}_{L}}{\sqrt{2}}=(\bar{x}\cos\tau
z+\bar{y}\sin\tau z)e^{i(kz-\omega t)},\\\
\mathbf{E}_{2}&=\frac{\mathbf{E}_{R}-\mathbf{E}_{L}}{i\sqrt{2}}=(-\bar{x}\sin\tau
z+\bar{y}\cos\tau z)e^{i(kz-\omega t)}.\end{split}$ (4)

Rotated in the same way, they must propagate at the same velocity. To see this
more clearly, we rewrite them in the following way,

$\begin{split}\mathbf{E}_{1}&=\bar{x}^{\prime}(z)\exp[i(kz-\omega t)],\\\
\mathbf{E}_{2}&=\bar{y}^{\prime}(z)\exp[i(kz-\omega t)],\end{split}$ (5)

where

$\begin{split}\bar{x}^{\prime}(z)&=\exp[-i(\bar{z}\cdot\mathbf{\Sigma})\tau
z]\bar{x},\\\ \bar{y}^{\prime}(z)&=\exp[-i(\bar{z}\cdot\mathbf{\Sigma})\tau
z]\bar{y},\end{split}$ (6)

$\exp[-i(\bar{z}\cdot\mathbf{\Sigma})\tau z]$ is the matrix of rotation about
the $z$-axis, $\bar{z}$ is the unit vector along the $z$-axis, and
$(\Sigma_{k})_{ij}=-i\epsilon_{ijk}$ with $\epsilon_{ijk}$ the Levi-Civitá
pseudotensor. Nevertheless, if $\mathbf{E}_{R}$ and $\mathbf{E}_{L}$ were
considered to propagate at different velocities, it would be impossible to
deduce from Eqs. (4) that $\mathbf{E}_{1}$ and $\mathbf{E}_{2}$ have the same
propagation velocity.

As a matter of fact, from Eqs. (4) one readily obtains

$\begin{split}\mathbf{E}_{R}&=\frac{1}{\sqrt{2}}(\mathbf{E}_{1}+i\mathbf{E}_{2}),\\\
\mathbf{E}_{L}&=\frac{1}{\sqrt{2}}(\mathbf{E}_{1}-i\mathbf{E}_{2}),\end{split}$
(7)

meaning that whether the RCP or the LCP wave is a coherent superposition of
the linearly polarized waves $\mathbf{E}_{1}$ and $\mathbf{E}_{2}$. Now that
these two linearly polarized waves propagate at the same velocity, the RCP and
LCP waves cannot propagate at different velocities. In other words, there is
no circular birefringence in the chiral medium. Then what is the propagation
velocity of plane waves in the chiral medium?

###  II.2 Propagation velocity of plane waves

When interpreting the RCP wave $\mathbf{E}_{R}$ in (2a) as propagating at
velocity $\frac{\omega}{k_{R}}$, one might imply that it is a superposition of
the following two orthogonal linearly polarized waves,

$\displaystyle\mathbf{E}_{1}^{R}$ $\displaystyle=\bar{x}\exp[i(k_{R}z-\omega
t)],$ (8a) $\displaystyle\mathbf{E}_{2}^{R}$
$\displaystyle=\bar{y}\exp[i(k_{R}z-\omega t)],$ (8b)

which propagate at velocity $\frac{\omega}{k_{R}}$. Meanwhile, when
interpreting the LCP wave $\mathbf{E}_{L}$ in (2b) as propagating at velocity
$\frac{\omega}{k_{L}}$, one might imply that it is a superposition of the
following two orthogonal linearly polarized waves,

$\begin{split}\mathbf{E}_{1}^{L}&=\bar{x}\exp[i(k_{L}z-\omega t)],\\\
\mathbf{E}_{2}^{L}&=\bar{y}\exp[i(k_{L}z-\omega t)],\end{split}$

which propagate at velocity $\frac{\omega}{k_{L}}$. But it is impossible for
one medium to transmit two sets of linearly polarized waves that, on the one
hand, have the same frequency and, on the other hand, propagate at different
velocities. The only possibility is that both of these sets of linearly
polarized waves do not exist. Indeed, the polarization of a linearly polarized
wave in the chiral medium cannot remain unchanged in transmission. It is
instead rotated with propagation as is shown by Eqs. (4) or (5).

As discussed before, the two linearly polarized waves $\mathbf{E}_{1}$ and
$\mathbf{E}_{2}$ propagate at the same velocity. Since the operator
$\exp[-i(\bar{z}\cdot\mathbf{\Sigma})\tau z]$ in (6) conveys the rotation of
their polarization states, the common phase factor $\exp[i(kz-\omega t)]$ in
(5) should be their propagation factor. It determines their phase velocity to
be $\frac{\omega}{k}$. More importantly, these two linearly polarized waves
are orthogonal to each other. They form a set of linearly polarized base modes
for plane waves propagating along the $z$-axis. Their linear combination gives
rise to a physically permissible wave,

$\mathbf{E}=\alpha_{1}\mathbf{E}_{1}+\alpha_{2}\mathbf{E}_{2}=\mathbf{a}(z)\exp[i(kz-\omega
t)],$ (9)

which is in general elliptically polarized, where

$\mathbf{a}(z)=\alpha_{1}\bar{x}^{\prime}(z)+\alpha_{2}\bar{y}^{\prime}(z)$
(10)

and the combination constants satisfy $|\alpha_{1}|^{2}+|\alpha_{2}|^{2}=1$.
In fact, as is shown by Eqs. (6), the unit vectors $\bar{x}^{\prime}$ and
$\bar{y}^{\prime}$ result from the same rotation of $\bar{x}$ and $\bar{y}$,
respectively, and are thus orthogonal to each other at the same propagation
distance $z$. As physically permissible polarization vectors, they serve as a
set of polarization bases [19] to expand the polarization vector $\mathbf{a}$
via Eq. (10). The expansion coefficients $\alpha_{1}$ and $\alpha_{2}$
constitute the well-known Jones vector [17, 18]
$\alpha=\bigg{(}\begin{array}[]{c}\alpha_{1}\\\
\alpha_{2}\end{array}\bigg{)}$. Upon substituting Eqs. (6) into Eq. (10), we
have

$\mathbf{a}(z)=\exp[-i(\bar{z}\cdot\mathbf{\Sigma})\tau
z](\alpha_{1}\bar{x}+\alpha_{2}\bar{y}).$ (11)

This indicates that the polarization state of elliptically polarized wave (9)
is rotated in the same way as those of linearly polarized waves (5),
irrespective of its Jones vector. Consequently, it propagates at the phase
velocity $\frac{\omega}{k}$.

###  II.3 Polarization vectors of RCP and LCP waves and rotation of
polarization bases

We are now clear that even though the RCP wave in the chiral medium is
correctly expressed by Eq. (2a), it does not propagate at the velocity
$\frac{\omega}{k_{R}}$. That is to say, $\exp[i(k_{R}z-\omega t)]$ is not its
propagation factor. This is equivalent to say that the RCP wave cannot be
resolved into the linearly polarized waves in (8) so that the constant unit
vectors $\bar{x}$ and $\bar{y}$ cannot be regarded as its polarization bases.
All of these means that $\frac{1}{\sqrt{2}}(\bar{x}+i\bar{y})$ is not the
polarization vector of the RCP wave. Similarly,
$\frac{1}{\sqrt{2}}(\bar{x}-i\bar{y})$ is not the polarization vector of the
LCP wave, either. Let us look at what the polarization vectors of the RCP and
LCP waves are.

The polarization vector of any elliptically polarized wave is given by Eq.
(10). It shows that the rotation of the polarization state comes only from the
rotation of the polarization bases $\bar{x}^{\prime}(z)$ and
$\bar{y}^{\prime}(z)$, without involving the change of the Jones vector
$\alpha$. When $\alpha=\frac{1}{\sqrt{2}}\bigg{(}\begin{array}[]{c}1\\\
i\end{array}\bigg{)}$, it becomes the polarization vector of the RCP wave and
is given by

$\mathbf{r}(z)=\frac{1}{\sqrt{2}}\exp[-i(\bar{z}\cdot\mathbf{\Sigma})\tau
z](\bar{x}+i\bar{y}).$ (12)

Likewise, when $\alpha=\frac{1}{\sqrt{2}}\bigg{(}\begin{array}[]{c}1\\\
-i\end{array}\bigg{)}$, it becomes the polarization vector of the LCP wave and
is given by

$\mathbf{l}(z)=\frac{1}{\sqrt{2}}\exp[-i(\bar{z}\cdot\mathbf{\Sigma})\tau
z](\bar{x}-i\bar{y}).$ (13)

Indeed, because $\bar{x}+i\bar{y}$ and $\bar{x}-i\bar{y}$ are eigen vectors of
matrix $\bar{z}\cdot\mathbf{\Sigma}$ with eigenvalues $+1$ and $-1$,
respectively, the above two equations can be re-expressed in the following
way,

$\begin{split}\mathbf{r}(z)&=\frac{1}{\sqrt{2}}(\bar{x}+i\bar{y})\exp(-i\tau
z),\\\ \mathbf{l}(z)&=\frac{1}{\sqrt{2}}(\bar{x}-i\bar{y})\exp(i\tau
z).\end{split}$ (14)

With the help of them, we can rewrite expressions (2) as

$\begin{split}\mathbf{E}_{R}(z,t)&=\mathbf{r}(z)\exp[i(kz-\omega t)],\\\
\mathbf{E}_{L}(z,t)&=\mathbf{l}(z)\exp[i(kz-\omega t)],\end{split}$ (15)

by virtue of Eqs. (3). From these discussions we conclude that the
polarization state of a circularly polarized wave, as a special case of
elliptically polarized wave (9), is not unchanged in transmission. It is
rotated with propagation in the same way as those of linearly polarized waves
(5). The only difference is that the result of rotation in such a case appears
as a $z$-dependent phase factor. That is to say, the $z$-dependent phase
factors in (14) convey a dependence of the polarization state on the
propagation distance. The phases acquired by the RCP and LCP waves over the
same propagation distance are the same in magnitude but are opposite in sign.
This may help to understand why the RCP and LCP waves were incorrectly
interpreted to propagate at different phase velocities in traditional
description.

It is important to point out that circular polarization vectors (12) and (13)
are also orthogonal to each other at the same propagation distance. They can
serve as a set of polarization bases, too. In terms of them, polarization
vector (10) can be expanded as follows,

$\mathbf{a}=\frac{1}{\sqrt{2}}(\alpha_{1}-i\alpha_{2})\mathbf{r}+\frac{1}{\sqrt{2}}(\alpha_{1}+i\alpha_{2})\mathbf{l}.$
(16)

It shows that the rotation of the circular polarization bases $\mathbf{r}$ and
$\mathbf{l}$ is responsible for the rotation of the polarization state of the
elliptically polarized wave when considered as a coherent superposition of the
RCP and LCP waves. In a word, even though the state of polarization described
by polarization vector (10) varies with the propagation distance $z$, the
Jones vector $\alpha$ does not. The variation of the state of polarization
amounts to the variation of the polarization bases $\bar{x}^{\prime}(z)$ and
$\bar{y}^{\prime}(z)$ with the propagation distance.

##  III Degrees of freedom for the state of polarization

Traditionally, the Jones vector was believed [18] to take the role of
describing the polarization of light. But the above discussions about the
optical activity suggest that this may not be the case. What completely
describes the polarization state of a plane wave can only be its polarization
vector. More importantly, as is shown by Eqs. (14), the change of polarization
state under some particular conditions appears as a phase factor of the
polarization vector. That is to say, polarization vectors that differ by phase
factors may not represent the same polarization state. This naturally raises
the question about the _nature_ of the polarization of light. It is generally
thought that the experimentally accessible quantities to characterize the
state of polarization are the Stokes parameters [15, 22, 23, 18, 24]. The
problem, however, is that what directly determines the Stokes parameters is
the Jones vector [17] rather than the polarization vector. Now that the Jones
vector is not able to completely describe the polarization, the Stokes
parameters are not able to completely characterize the polarization, either.
To truly understand the nature of the polarization, it is crucial to figure
out the degrees of freedom that determine the state of polarization. Let us
first make use of the polarization vector of a plane wave in a dielectric
medium, which is achiral, to analyze how many degrees of freedom are needed to
determine a state of polarization.

###  III.1 Number of degrees of freedom

For the sake of clarity, the plane wave is assumed to propagate in an
arbitrary direction denoted by $\mathbf{w}$, the unit wavevector. The electric
field of it can be written in the form:

$\mathbf{E}(\mathbf{x},t)=\mathbf{a}E\exp[i(k\mathbf{w}\cdot\mathbf{x}-\omega
t)],$ (17)

where $\mathbf{a}$ is the polarization vector obeying the normalization
condition

$\mathbf{a}^{*}\cdot\mathbf{a}=1,$ (18)

$E$ is the amplitude, and $k$ is the wave number. What should be kept in mind
is that the polarization vector has nothing to do with the phase coming from
the propagation factor $\exp[i(k\mathbf{w}\cdot\mathbf{x}-\omega t)]$. It has
three Cartesian components,

$\mathbf{a}=a_{x}\bar{x}+a_{y}\bar{y}+a_{z}\bar{z}\equiv\Bigg{(}\begin{array}[]{c}a_{x}\\\
a_{y}\\\ a_{z}\end{array}\Bigg{)}.$

Each of them is a complex-valued number. So after normalization condition (18)
is taken into consideration, there are five independent real-valued parameters
or, equivalently, five degrees of freedom to determine the polarization
vector. However, the polarization vector has to be perpendicular to the
propagation direction $\mathbf{w}$,

$\mathbf{w}\cdot\mathbf{a}=0,$

in accordance with the Maxwell’s equation $\nabla\cdot\mathbf{E}=0$. This
means that the two independent real-valued parameters specifying the
propagation direction $\mathbf{w}$ are two degrees of freedom to determine a
state of polarization. As a result, apart from $\mathbf{w}$, three degrees of
freedom are still needed.

###  III.2 Stokes parameters contain only two degrees of freedom

Now we pay attention to the Stokes parameters. As mentioned above, it is
impossible to determine the Stokes parameters directly from the polarization
vector. To do this, one has to introduce the Jones vector. Let be $\mathbf{u}$
and $\mathbf{v}$ two mutually-perpendicular unit vectors that, with
$\mathbf{w}$, form a right-handed Cartesian coordinate system,

$\mathbf{u}\cdot\mathbf{u}=\mathbf{v}\cdot\mathbf{v}=1,\quad\mathbf{u}\cdot\mathbf{v}=0,\quad\mathbf{u}\times\mathbf{v}=\mathbf{w}.$
(19)

Being perpendicular to $\mathbf{w}$, they represent two orthogonal
polarization states that are physically permissible in the achiral medium.
When they are taken as the polarization bases to expand the polarization
vector,

$\mathbf{a}=\alpha_{1}\mathbf{u}+\alpha_{2}\mathbf{v},$ (20)

the expansion coefficients constitute the Jones vector
$\alpha=\bigg{(}\begin{array}[]{c}\alpha_{1}\\\
\alpha_{2}\end{array}\bigg{)}$, which is normalized

$\alpha^{\dagger}\alpha=1$ (21)

in accordance with Eq. (18). Upon expressing the polarization bases as three-
component column matrices $\mathbf{u}=\Bigg{(}\begin{array}[]{c}u_{x}\\\
u_{y}\\\ u_{z}\end{array}\Bigg{)}$ and
$\mathbf{v}=\Bigg{(}\begin{array}[]{c}v_{x}\\\ v_{y}\\\
v_{z}\end{array}\Bigg{)},$ we can convert expansion (20) into

$\mathbf{a}=\varpi\alpha,$ (22)

where $\varpi=(\begin{array}[]{lr}\mathbf{u}&\mathbf{v}\end{array})$ is a
$3\times 2$ matrix. The matrix $\varpi$ satisfies

$\varpi^{\dagger}\varpi=I_{2}$ (23)

by virtue of Eqs. (19), where $I_{2}$ is the unit $2\times 2$ matrix.
Multiplying (22) by $\varpi^{\dagger}$ on the left and making use of (23), we
get

$\alpha=\varpi^{\dagger}\mathbf{a}.$ (24)

In terms of the Jones vector, the Stokes parameters are expressed as

$s_{i}=\alpha^{\dagger}\hat{\sigma}_{i}\alpha,\quad i=1,2,3,$ (25)

where

$\hat{\sigma}_{1}=\bigg{(}\begin{array}[]{cc}1&0\\\
0&-1\end{array}\bigg{)},\quad\hat{\sigma}_{2}=\bigg{(}\begin{array}[]{cc}0&1\\\
1&0\end{array}\bigg{)},\quad\hat{\sigma}_{3}=\bigg{(}\begin{array}[]{cc}0&-i\\\
i&0\end{array}\bigg{)}$ (26)

are the Pauli matrices.

Of course, if one chooses circular polarization bases [19], which are given by

$\displaystyle\mathbf{R}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\mathbf{u}+i\mathbf{v}),$
$\displaystyle\mathbf{L}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\mathbf{u}-i\mathbf{v}),$

the polarization vector can be expanded as follows,

$\mathbf{a}=\alpha_{R}\mathbf{R}+\alpha_{L}\mathbf{L}\equiv\varpi^{c}\alpha^{c},$

where $\alpha_{R}=\frac{1}{\sqrt{2}}(\alpha_{1}-i\alpha_{2})$,
$\alpha_{L}=\frac{1}{\sqrt{2}}(\alpha_{1}+i\alpha_{2})$,
$\varpi^{c}=(\begin{array}[]{lr}\mathbf{R}&\mathbf{L}\end{array})$, and
$\alpha^{c}=\bigg{(}\begin{array}[]{c}\alpha_{R}\\\
\alpha_{L}\end{array}\bigg{)}$. In this case, the Stokes parameters (25) can
be rewritten as

$s_{i}=\alpha^{c{\dagger}}\hat{\sigma}^{c}_{i}\alpha^{c},$

where

$\hat{\sigma}^{c}_{1}=\bigg{(}\begin{array}[]{cc}0&1\\\
1&0\end{array}\bigg{)},\quad\hat{\sigma}^{c}_{2}=\bigg{(}\begin{array}[]{cc}0&-i\\\
i&0\end{array}\bigg{)},\quad\hat{\sigma}^{c}_{3}=\bigg{(}\begin{array}[]{cc}1&0\\\
0&-1\end{array}\bigg{)}.$

The circular polarization bases $\varpi^{c}$ are related to the linear
polarization bases $\varpi$ through

$\varpi^{c}=\varpi M^{\dagger},$ (27)

where $M=\frac{1}{\sqrt{2}}\bigg{(}\begin{array}[]{cc}1&-i\\\
1&i\end{array}\bigg{)},$ so that $\alpha^{c}=M\alpha$ and
$\hat{\sigma}^{c}_{i}=M\hat{\sigma}_{i}M^{\dagger}$. The transition from the
linear polarization bases to the circular polarization bases does not affect
the Stokes parameters. In this paper, we will only use the linear polarization
bases.

According to normalization condition (21), the Stokes parameters satisfy

$s_{1}^{2}+s_{2}^{2}+s_{3}^{2}=1,$

meaning that only two of the three Stokes parameters are independent. In other
words, only two degrees of freedom are contained in the Stokes parameters. We
are thus convinced that the Stokes parameters, together with the propagation
direction $\mathbf{w}$, are not enough to completely determine a state of
polarization. One additional degree of freedom is needed.

###  III.3 The fifth degree of freedom and local coordinate system

To find out the additional degree of freedom, we reexamine the Jones vector
that fully determines the Stokes parameters. It is noticed that the transverse
axes that are taken to introduce the Jones vector via (24) are indeterminate
to the extent that a rotation about the propagation direction can be performed
[25]. By this it is meant that the Jones vector is an entity in association
with the transverse axes. For a given polarization vector, the Jones vector
associated with different transverse axes should be different. Denote by
$\varpi^{\prime}=(\begin{array}[]{cc}\mathbf{u}^{\prime}&\mathbf{v}^{\prime}\end{array})$
the new transverse axes that result from a rotation of the old ones, $\varpi$,
by an angle $\phi$,

$\varpi^{\prime}=\exp[-i(\mathbf{w}\cdot\mathbf{\Sigma})\phi]\varpi.$ (28)

Taking the primed unit vectors $\mathbf{u}^{\prime}$ and $\mathbf{v}^{\prime}$
as new polarization bases, the same polarization vector that is expanded in
terms of the old polarization bases $\varpi$ via (22) can be expanded as

$\mathbf{a}=\alpha^{\prime}_{1}\mathbf{u}^{\prime}+\alpha^{\prime}_{2}\mathbf{v}^{\prime}=\varpi^{\prime}\alpha^{\prime},$

where the expansion coefficients $\alpha^{\prime}_{1}$ and
$\alpha^{\prime}_{2}$ constitute the new Jones vector
$\alpha^{\prime}=\bigg{(}\begin{array}[]{c}\alpha^{\prime}_{1}\\\
\alpha^{\prime}_{2}\end{array}\bigg{)}$. Since the new polarization bases also
satisfies $\varpi^{\prime{\dagger}}\varpi^{\prime}=I_{2}$, we multiply the
above equation by $\varpi^{\prime{\dagger}}$ on the left and take this
relation into account to give

$\alpha^{\prime}=\varpi^{\prime{\dagger}}\mathbf{a}.$ (29)

Upon substituting Eqs. (28) and (22), we finally get

$\alpha^{\prime}=\exp(i\hat{\sigma}_{3}\phi)\alpha,$ (30)

where the equality

$\hat{\sigma}_{3}=\varpi^{\dagger}(\mathbf{w}\cdot\mathbf{\Sigma})\varpi$

has been used. Eq. (30) constitutes the transformation of the Jones vector of
the given polarization vector $\mathbf{a}$ under the transformation (28) of
the transverse axes. It indicates that the new Jones vector cannot be the same
as the old one unless $\phi=0$ or $\phi=2\pi$. This suggests that the Jones
vector as a two-component entity is not defined in the same coordinate system
as the polarization vector. If the polarization vector is said to be defined
in the laboratory coordinate system, the Jones vector cannot be.

To discern the coordinate system in which the Jones vector is defined, we turn
our attention back to the Stokes parameters. It is usually thought [17, 19]
that the Stokes parameters are described in an abstract three-dimensional
space, constituting a vector in that space. However, since the new
polarization bases $\varpi^{\prime}$ are on the same footing as the old ones
$\varpi$, it follows from Eq. (25) that the Stokes parameters determined by
the new Jones vector $\alpha^{\prime}$ are given by

$s^{\prime}_{i}=\alpha^{\prime{\dagger}}\hat{\sigma}_{i}\alpha^{\prime},\quad
i=1,2,3.$ (31)

Now that $\alpha^{\prime}$ is different from $\alpha$, the new Stokes
parameters in (31) are in general not the same as the old ones in (25). In
fact, substituting (30) into (31) and making use of (25), we get

$\begin{split}s^{\prime}_{1}&=s_{1}\cos 2\phi+s_{2}\sin 2\phi,\\\
s^{\prime}_{2}&=-s_{1}\sin 2\phi+s_{2}\cos 2\phi,\\\
s^{\prime}_{3}&=s_{3}.\end{split}$ (32)

Because it is impossible for a given state of polarization to have different
sets of Stokes parameters in one and the same space, the three-dimensional
space in which the Stokes parameters are described cannot be an abstract
space. The Stokes parameters must be defined in some concrete coordinate
system. Different sets of Stokes parameters are defined in different
coordinate systems. Considering that they are solely determined by the Jones
vector, the Stokes parameters should be defined in the same coordinate system
as the Jones vector. What is more important is that transformation (30) of the
Jones vector has a one-to-one correspondence with transformation (28) of the
transverse axes. In particular, when $\phi=\pi$ (or
$\varpi^{\prime}=-\varpi$), one has $\alpha^{\prime}=-\alpha$. And when
$\phi=2\pi$ (or $\varpi^{\prime}=\varpi$), one has $\alpha^{\prime}=\alpha$.
Thus, we have reason to believe that the Jones vector as well as the Stokes
parameters is defined in the coordinate system formed by the associated
transverse axes and the propagation direction $\mathbf{w}$. This is, of
course, not the laboratory coordinate system in which the polarization vector
is defined. It is a local coordinate system dependent on the propagation
direction. Apparently, if the local coordinate system is not specified, it
will be physically meaningless to talk about a Jones vector or a set of Stokes
parameters. From this one may infer that the degree of freedom that we are
searching for should have something to do with the local coordinate system in
which the Jones vector and the Stokes parameters are defined.

Mathematically, the Jones vector is related to the polarization vector by Eq.
(24). If the Jones vector is defined in the local coordinate system that
depends on the propagation direction, then Eq. (24) means that the mapping
matrix $\varpi^{\dagger}$ is enough to represent the local coordinate system
in which it is defined. This is indeed the case. As is indicated by Eqs. (19),
the transverse axes $\mathbf{u}$ and $\mathbf{v}$ must be connected with some
particular propagation direction $\mathbf{w}$. So the matrix
$\varpi^{\dagger}$ that consists of $\mathbf{u}$ and $\mathbf{v}$ already
conveys the information of $\mathbf{w}$ in an appropriate way. But on the
other hand, Eq. (28) shows that given $\mathbf{w}$, different local coordinate
systems are related to one another by a rotation about it. That is to say, the
rotation angle $\phi$ in (28) is the degree of freedom to distinguish
different local coordinate systems that all depend on the same $\mathbf{w}$.
Now that the Stokes parameters are quantities in the local coordinate system,
for the Stokes parameters to be physically meaningful, the local coordinate
system has to be specified unambiguously. The rotation angle $\phi$ is
therefore the fifth degree of freedom to determine the state of polarization
of a plane wave. It combines with the propagation direction $\mathbf{w}$ to
specify the local coordinate system. It is noted, by the way, that the
rotation of the local coordinate system, Eq. (28), is not to be confused with
the transition of the polarization bases, Eq. (27), which does not involve the
change of the local coordinate system.

In concluding this section we summarize our main results. In order to
completely determine the state of polarization of a plane wave, two different
kinds of degrees of freedom are needed. One is the Stokes parameters that
contain two degrees of freedom. The other is the degrees of freedom to specify
the local coordinate system in which the Stokes parameters are defined,
including the propagation direction and an angle of rotation about it. The
reason for the Stokes parameters not to be able to completely characterize the
polarization is that they are quantities in the local coordinate system
whereas the polarization is a phenomenon in the laboratory coordinate system.
Let us consider what on earth these experimentally accessible quantities
really mean.

##  IV Quasi-spin in local coordinate system

It is well known in quantum mechanics that the Pauli matrices are
representative operators for the electron spin. The wavefunction to describe
the state of electron spin is a two-component spinor. So an analogy with the
spin of the electron suggests that the Stokes parameters in (25) correspond to
an observable the representative operators for which are the Pauli matrices
(26). They are expectation values of these operators in a state described by
the wavefunction, the Jones vector $\alpha$. An important property of the
Pauli matrices is that they satisfy the commutation relation of the angular
momentum,

$[\hat{\sigma}_{i},\hat{\sigma}_{j}]=2i\sum_{k}\epsilon_{ijk}\hat{\sigma}_{k},\quad
i,j,k=1,2,3,$ (33)

except for a factor two. In addition, as discussed above, the Stokes
parameters constitute a three-dimensional unit vector in the associated local
coordinate system. In fact, transformations (32) indicate that $s_{3}$ is the
component of the vector along the propagation direction $\mathbf{w}$ and,
hence, $s_{1}$ and $s_{2}$ are the components along the transverse axes
$\mathbf{u}$ and $\mathbf{v}$, respectively,

$\mathbf{P}=s_{1}\mathbf{u}+s_{2}\mathbf{v}+s_{3}\mathbf{w}.$ (34)

According to Merzbacher [19], we will call $\mathbf{P}$ the Poincaré vector.
Substituting Eq. (25) into Eq. (34), we have

$\mathbf{P}=\alpha^{\dagger}\hat{\boldsymbol{\sigma}}\alpha,$

where

$\hat{\boldsymbol{\sigma}}=\hat{\sigma}_{1}\mathbf{u}+\hat{\sigma}_{2}\mathbf{v}+\hat{\sigma}_{3}\mathbf{w}.$
(35)

Eq. (35) shows that similar to the spin of the electron, the observable
represented by the Pauli matrices here is also a vector. But unexpectedly, its
quantization axis is with respect to the local coordinate system. That is to
say, it is quantized with respect to the local coordinate system. In this
sense we say that it is the quantum-mechanical property of light in the local
coordinate system. For these reasons, we will refer to this observable as the
quasi-spin of the photon.

It is not difficult to prove [17] that the wavefunction of the quasi-spin, the
Jones vector $\alpha$, is an eigenfunction of the unitary unimodular matrix
$\mathbf{P}\cdot\hat{\boldsymbol{\sigma}}$:

$\mathbf{P}\cdot\hat{\boldsymbol{\sigma}}\alpha=\alpha.$ (36)

Hence, the Poincaré vector $\mathbf{P}$ may legitimately be said to point in
the direction of the quasi-spin in the local coordinate system. In other
words, the Poincaré vector, or equivalently, the Stokes parameters, takes the
role to characterize the state of quasi-spin in the local coordinate system.
This, in turn, shows that Jones vectors that differ by phase factors describe
the same state of quasi-spin, the same as is demonstrated by the bilinear
dependence of the Stokes parameters (25) on the Jones vector.

Algebraic relation (33) means that the operator $\hat{\boldsymbol{\sigma}}$
for the quasi-spin is the generator of rotation in the _local coordinate
system_. This is to say that the operator
$\exp(-\frac{i}{2}\mathbf{n}\cdot\boldsymbol{\sigma}\chi)$ induces a rotation
about the direction $\mathbf{n}$ by an angle $\chi$, where
$\mathbf{n}=n_{1}\mathbf{u}+n_{2}\mathbf{v}+n_{3}\mathbf{w}$ is a unit vector
in the local coordinate system. It changes the quasi-spin state $\alpha$ into

$\alpha^{\prime\prime}=\exp\Big{(}-\frac{i}{2}\mathbf{n}\cdot\boldsymbol{\sigma}\chi\Big{)}\alpha.$
(37)

So the Jones vector behaves in the local coordinate system as a spinor [24,
19, 26, 27]. After the rotation, the Poincaré vector of the quasi-spin state
$\alpha^{\prime\prime}$ is given by

$\mathbf{P}^{\prime\prime}=\alpha^{\prime\prime{\dagger}}\hat{\boldsymbol{\sigma}}\alpha^{\prime\prime}=s^{\prime\prime}_{1}\mathbf{u}+s^{\prime\prime}_{2}\mathbf{v}+s^{\prime\prime}_{3}\mathbf{w},$

where
$s^{\prime\prime}_{i}=\alpha^{\prime\prime{\dagger}}\hat{\sigma}_{i}\alpha^{\prime\prime}$.
It is easy [17, 28] to prove that if letting
$\mathbf{P}=\Bigg{(}\begin{array}[]{c}s_{1}\\\ s_{2}\\\
s_{3}\end{array}\Bigg{)}$ and
$\mathbf{P}^{\prime\prime}=\Bigg{(}\begin{array}[]{c}s^{\prime\prime}_{1}\\\
s^{\prime\prime}_{2}\\\ s^{\prime\prime}_{3}\end{array}\Bigg{)}$, one will
have

$\mathbf{P}^{\prime\prime}=\exp[-i(\mathbf{n}\cdot\mathbf{\Sigma})\chi]\mathbf{P},$
(38)

showing that the Stokes parameters indeed transform as the Cartesian
components of a vector under rotations in the local coordinate system.

In a word, as revealed by the analyses of the phenomenon of optical activity
that we conducted in Section II, the traditional theory for describing the
polarization of light is incomplete. What the Jones vector truly describes is
not the polarization. Instead, it describes the quantum-mechanical property of
light in the local coordinate system, the quasi-spin. Such a property has
profound implications for the nature of the polarization as we will show
below.

##  V Physical meaning of polarization

The polarization vector for the state of polarization is related to the Jones
vector for the state of quasi-spin by Eq. (22). What is interesting is, as we
have seen before, that the matrix $\varpi$ in Eq. (22) elegantly represents
the local coordinate system with respect to which the quasi-spin is quantized.
Let us make use of the plane wave (17) in the achiral medium to examine in
detail how the state of polarization depends on the state of quasi-spin in the
local coordinate system.

In the first place, we examine the dependence of the state of polarization on
the state of quasi-spin in a fixed local coordinate system. Assume that the
local coordinate system $\varpi$ involved in Eqs. (22)-(24) is fixed.
Substituting Eq. (24) into Eq. (22) and considering the arbitrariness of
$\mathbf{a}$, we get

$\varpi\varpi^{\dagger}=I_{3},$ (39)

where $I_{3}$ is the unit $3\times 3$ matrix. Mathematically, Eq. (39) in
combination with Eq. (23) means that the matrix $\varpi$ is quasi-unitary
[29]. $\varpi^{\dagger}$ is the Moore-Penrose pseudo inverse of $\varpi$, and
vice versa. The quasi-unitarity of $\varpi$ guarantees a one-to-one
correspondence between the Jones vector $\alpha$ and the polarization vector
$\mathbf{a}$ via Eq. (22) or (24). Hence, to each quasi-spin state $\alpha$ in
the fixed local coordinate system there corresponds a unique polarization
state $\mathbf{a}$ in the laboratory coordinate system. In this case, the
Stokes parameters (25) that characterize the state of quasi-spin can be used
to differentiate between different states of polarization. Such a
correspondence provides a mechanism to change the state of polarization, which
is to vary the state of quasi-spin in a fixed local coordinate system. This is
the traditional mechanism that can be interpreted in terms of the variation of
the Jones vector [17]. It should be pointed out that if the $z$-axis of the
laboratory coordinate system is taken as the propagation direction, the $x$\-
and $y$-axes can also be chosen as the transverse axes of the local coordinate
system. This is the local coordinate system that is commonly chosen in the
literature [18] to define the Jones vector. The measurement of the
corresponding Stokes parameters therefore should be understood to be done in
so chosen local coordinate system. But such a choice makes it difficult to
distinguish the local coordinate system for the quasi-spin from the laboratory
coordinate system for the polarization.

In the second place, we examine the dependence of the state of polarization on
the local coordinate system in which the state of quasi-spin is fixed. To this
end, we consider a polarization state the Jones vector of which is the same as
that of polarization state (22) but is defined in the primed local coordinate
system $\varpi^{\prime}$,

$\mathbf{a}^{\prime}=\varpi^{\prime}\alpha.$ (40)

According to Eq. (28), polarization state (40) is different from polarization
state (22) unless $\phi=0$ or $\phi=2\pi$. The rotation angle $\phi$ for the
local coordinate system is now the degree of freedom to differentiate between
different polarization states. The Stokes parameters can no longer be able to
play the role.

We stress that the Stokes parameters constitute a vector, the Poincaré vector
(34), in the local coordinate system. But it should be noted that
$\mathbf{u}$, $\mathbf{v}$, and $\mathbf{w}$ for the Cartesian axes of the
local coordinate system are all unit vectors in the laboratory coordinate
system. In this regard, the Poincaré vector can also be “viewed” as a vector
in the laboratory coordinate system. Even so, it is not able to completely
characterize the state of polarization, the same as the Stokes parameters. We
have seen in Part III.3 that a given state of polarization corresponds to
different states of quasi-spin in different local coordinate systems [30] as
is shown by Eqs. (24) and (29). The Poincaré vector in the unprimed local
coordinate system $\varpi$ is given by expression (34). Accordingly, the
Poincaré vector in the primed local coordinate system $\varpi^{\prime}$ is
given by

$\mathbf{P}^{\prime}=s^{\prime}_{1}\mathbf{u}^{\prime}+s^{\prime}_{2}\mathbf{v}^{\prime}+s^{\prime}_{3}\mathbf{w},$

where $s^{\prime}_{i}$ ($i=1,2,3$) are defined by (31). Upon substituting Eqs.
(32) and taking Eq. (28) into consideration, it is not difficult to find

$\mathbf{P}^{\prime}=\exp[i(\mathbf{w}\cdot\mathbf{\Sigma})\phi]\mathbf{P},$

showing that the Poincaré vectors in different local coordinate systems, when
viewed in the laboratory coordinate system, are in general not the same. This
proves our assertion. Ultimately, as is shown by Eq. (36), what the Poincaré
vector characterizes is the state of quasi-spin in the local coordinate
system.

To conclude, the polarization of light described by the polarization vector is
the reflection of the quasi-spin of the photon in its local coordinate system.
It not only has the property of the quasi-spin but also conveys the
information of the local coordinate system in which the quasi-spin is
characterized. The reason for the traditional theory for describing the
polarization of light to be incomplete is that it simply identifies the
polarization in the laboratory coordinate system with the quasi-spin in some
specific local coordinate system [28]. This is why it could not correctly
describe the phenomenon of optical activity in the chiral medium. The
dependence of the polarization on the local coordinate system provides a new
mechanism to change the polarization. This is to vary the local coordinate
system with the state of quasi-spin remaining fixed in it. We will see in the
next section that it is this newly identified mechanism that underlies the
variation of the polarization bases in the chiral medium.

##  VI A new mechanism to change the state of polarization

To better understand the new mechanism for changing the polarization, it is
helpful to first interpret the traditional mechanism in terms of the quasi-
spin. As discussed above, the state of quasi-spin in a fixed local coordinate
system has a one-to-one correspondence with the state of polarization in the
laboratory coordinate system. So varying the state of quasi-spin in a fixed
local coordinate system is able to change the state of polarization. Such a
variation can be mathematically expressed as a rotation of the Jones vector as
is shown by (37). Suppose that the marix $\varpi^{\dagger}$ in Eq. (24)
represents the fixed local coordinate system. If the quasi-spin state
$\alpha^{\prime\prime}$ is different from the quasi-spin state $\alpha$, the
polarization state

$\mathbf{a}^{\prime\prime}=\varpi\alpha^{\prime\prime}$

will be different from polarization state (22). This mechanism can be
experimentally implemented by applying waveplates. The fast and slow axes of a
waveplate take the role of choosing the transverse axes of the local
coordinate system in which the Jones vector is varied [18]. In that case, Eq.
(24) can be interpreted as mapping the polarization vector $\mathbf{a}$ onto
the Jones vector $\alpha$ in the chosen local coordinate system that is
represented by $\varpi^{\dagger}$. Upon substituting Eq. (37) into the above
equation and considering Eq. (24), one has

$\mathbf{a}^{\prime\prime}=\varpi\exp\Big{(}-\frac{i}{2}\mathbf{n}\cdot\hat{\boldsymbol{\sigma}}\chi\Big{)}\varpi^{\dagger}\mathbf{a}.$
(41)

Now that the quasi-spin is a quantum-mechanical property with respect to the
local coordinate system, varying the local coordinate system with the state of
quasi-spin remaining fixed in it is also able to change the state of
polarization. For instance, if rotating the local coordinate system about the
propagation direction without varying the state of quasi-spin in it, one will
change the state of polarization. This is demonstrated by the difference
between Eqs. (40) and (22). In fact, substituting Eq. (28) into Eq. (40) and
taking Eq. (22) into consideration, one gets

$\mathbf{a}^{\prime}=\exp[-i(\mathbf{w}\cdot\mathbf{\Sigma})\phi]\mathbf{a},$
(42)

which shows that the rotation angle $\phi$ here is a continuous variable to
determine the state of polarization. What we encountered in Section II is
precisely such a physical process. Indeed, from Eq. (11) we have

$\mathbf{a}(0)=\alpha_{1}\bar{x}+\alpha_{2}\bar{y}.$

As a result, we can rewrite Eq. (11) as follows,

$\mathbf{a}(z)=\exp[-i(\bar{z}\cdot\mathbf{\Sigma})\tau z]\mathbf{a}(0).$

This is to say that the rotation of the polarization bases in the chiral
medium is the reflection of the rotation of the local coordinate system about
the propagation direction. It is particularly pointed out that since the
rotation angle $\tau z$ there is proportional to the propagation distance, the
state of polarization at $z=z_{1}$ is different from the state of polarization
at a different propagation distance, say, $z=z_{2}$, even when
$\tau(z_{2}-z_{1})=2m\pi$ with $m$ a non-zero integer. More interestingly, a
helically-coiled fiber that is free of linear birefringence not only rotates
the transverse axes of the local coordinate system but also changes the
propagation direction [3, 4, 5, 6] if a locally-paraxial approximation is made
to the wave that passes through it.

The difference between the new and the traditional mechanisms can be further
explained as follows. It is seen from Eq. (41) that the traditional mechanism
is mathematically expressed as an SU(2) rotation of the Jones vector within a
local coordinate system. The corresponding Poincaré vector undergoes an SO(3)
rotation in the same local coordinate system, as is indicated by Eq. (38).
However, as is shown by Eq. (42), the new mechanism leads to an SO(3) rotation
of the polarization vector in the laboratory coordinate system. What is
important is that the Poincaré vector in this case cannot always embody such a
rotation even though it can be viewed as a vector in the laboratory coordinate
system. To illustrate this more clearly, let the state of quasi-spin be an
eigenstate of $\hat{\sigma}_{3}$, say,
$\alpha=\frac{1}{\sqrt{2}}\bigg{(}\begin{array}[]{c}1\\\
i\end{array}\bigg{)}$. As we have just seen, the polarization state,
$\mathbf{a}^{\prime}=\frac{1}{\sqrt{2}}(\mathbf{u}^{\prime}+i\mathbf{v}^{\prime})$,
in association with the primed local coordinate system $\varpi^{\prime}$ is
different from the polarization state,
$\mathbf{a}=\frac{1}{\sqrt{2}}(\mathbf{u}+i\mathbf{v})$, in association with
the unprimed local coordinate system $\varpi$. As a matter of fact, since
$\mathbf{u}+i\mathbf{v}$ is an eigen vector of the operator
$\mathbf{w}\cdot\mathbf{\Sigma}$ with the eigenvalue $+1$, it follows from Eq.
(42) that $\mathbf{a}^{\prime}$ is different from $\mathbf{a}$ by a phase
factor,

$\mathbf{a}^{\prime}=\exp(-i\phi)\mathbf{a}.$

But the Poincaré vectors in both local coordinate systems are the same and
take the form of $\mathbf{w}$. This again shows that the Poincaré vector is
not able to completely characterize the polarization of a plane light wave.
The key point here is that the Poincaré vector, when viewed in the laboratory
coordinate system, generally has a non-vanishing longitudinal component, in
sharp contrast to the polarization vector that has no longitudinal component.

##  VII Conclusions and Remarks

In conclusion, realizing that the traditional description of the polarization
of light is incomplete, we reexamined the nature of the polarization from the
point of view of measurement. We found that in order to completely determine
the polarization state of a plane light wave, two different kinds of degrees
of freedom are needed. One is the Stokes parameters. The other is the degrees
of freedom to specify the local coordinate system in which the Stokes
parameters are defined. This is because described by the polarization vector
in the laboratory coordinate system, the polarization of a light wave is the
reflection of its quantum-mechanical property, the quasi-spin, in the
momentum-dependent local coordinate system. The wavefunction of the quasi-spin
is the Jones vector. As is indicated by (24), the Jones vector is the
projection of the polarization vector onto the local coordinate system
represented by the matrix $\varpi^{\dagger}$. It appears as a spinor in the
local coordinate system. The representative operator for the quasi-spin is
given by (35). The SU(2) algebra (33) makes the Stokes parameters (25) form a
vector, the Poincaré vector (34), in the local coordinate system. The Poincaré
vector, or equivalently, the Stokes parameters, characterizes the state of
quasi-spin in accordance with (36).

On the one hand, the state of quasi-spin in a definite local coordinate system
has a one-to-one correspondence with the state of polarization in the
laboratory coordinate system. In this case, the Stokes parameters can be used
to differentiate between different states of polarization. This is what is
done in traditional description. On the other hand, a definite state of quasi-
spin in different local coordinate systems corresponds to different states of
polarization. The Stokes parameters in such a case can no longer be used to
distinguish different states of polarization. Consequently, there are two
distinct mechanisms to change the state of polarization without affecting the
intensity. One is to change the state of quasi-spin, or the Stokes parameters,
in a fixed local coordinate system. The other is to change the local
coordinate system with the Stokes parameters remaining fixed in it. The former
is the traditional mechanism that can be expressed as an SU(2) rotation of the
Jones vector. The latter, which is newly identified here, is found to be
responsible for the optical activity in the chiral medium.

In a word, the Stokes parameters by themselves are not able to completely
characterize the polarization of a pure state in classical optics. This result
provides a deeper insight into the nature of optical polarization. It is
closely related to the “hidden polarization” [31] that was conjectured a long
time ago [32, 33]. It should be pointed out that the quasi-spin of the photon
is not to be confused with the spin angular momentum that combines the orbital
angular momentum to give the total angular momentum [34, 35].

## Acknowledgments

This work was supported in part by National Natural Science Foundation of
China under Grant No. 11974251.

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