Geometric phases for finite-dimensional systems -- the roles of Bargmann Invariants, Null Phase Curves and the Schwinger Majorana SU(2) framework

TL;DR

This paper investigates the properties of Bargmann Invariants and Null Phase Curves in finite-dimensional quantum systems, utilizing the Majorana theorem and Schwinger framework to develop new operator methods for understanding geometric phases.

Contribution

It combines the Majorana theorem with Schwinger oscillator methods to create efficient operator-based approaches for analyzing geometric phases in finite-dimensional systems.

Findings

Analysis of algebraic properties of BI using group theory

Extension of BI-geometric phase connection with NPCs

Proposals for new experiments in geometric phase studies

Abstract

We present a study of the properties of Bargmann Invariants (BI) and Null Phase Curves (NPC) in the theory of the geometric phase for finite dimensional systems. A recent suggestion to exploit the Majorana theorem on symmetric SU(2) multispinors is combined with the Schwinger oscillator operator construction to develop efficient operator based methods to handle these problems. The BI is described using intrinsic unitary invariant angle parameters, whose algebraic properties as functions of Hilbert space dimension are analysed using elegant group theoretic methods. The BI-geometric phase connection, extended by the use of NPC's, is explored in detail, and interesting new experiments in this subject are pointed out.

Equations

$$\mathcal{B}$$

$$\mathcal{R}$$

$$\pi :\mathcal{B}\to \mathcal{R}:\psi \to \rho (\psi ).$$

$${\Delta }_3({\psi }_1,{\psi }_2,{\psi }_3)=({\psi }_1,{\psi }_2)({\psi }_2,{\psi }_3)({\psi }_3,{\psi }_1),$$

$${\Delta }_3({\psi }_1,{\psi }_2,{\psi }_3)=Tr(\rho ({\psi }_1)\rho ({\psi }_2)\rho ({\psi }_3)).$$

$$({\psi }_1,{\psi }_2)=\cos \frac{1}{2}{\theta }_0,0<{\theta }_0<\pi .$$

$$C_{\mathrm{geo},12}$$

$${\mathcal{C}}_{\mathrm{geo},12}$$

$$C:\rho ({\psi }_1)\to \rho ({\psi }_1):$$

$$C=C_{\mathrm{geo},12}\cup C_{\mathrm{geo},23}\cup C_{\mathrm{geo},31}$$

$${\phi }_{\mathrm{geom}}[C_{\mathrm{geo},12}\cup C_{\mathrm{geo},23}\cup C_{\mathrm{geo},31}]=-\arg {\Delta }_3({\psi }_1,{\psi }_2,{\psi }_3).$$

$${\psi }_j^{\prime }=U{\psi }_j,j=1,2,3:{\Delta }_3({\psi }_1^{\prime },{\psi }_2^{\prime },{\psi }_3^{\prime })={\Delta }_3({\psi }_1,{\psi }_2,{\psi }_3).$$

$$jk=12,23,31:({\psi }_j,{\psi }_k)=e^{i{\phi }_{jk}}\cos \frac{1}{2}{\theta }_{jk},$$

$$0<{\theta }_{jk}<\pi ,0\le {\phi }_{jk}<2\pi .$$

$${\psi }_j^{\prime }=e^{i{\alpha }_j}{\psi }_j:{\phi }_{jk}^{\prime }={\phi }_{jk}-{\alpha }_j+{\alpha }_k.$$

$$-\arg {\Delta }_3({\psi }_1,{\psi }_2,{\psi }_3)=-({\phi }_{12}+{\phi }_{23}+{\phi }_{31})=a\; geometric\; phase.$$

$${\psi }_3$$

$${\phi }_g$$

$$S_{12}$$

$$\psi_{1}=\left(\begin{array}[]{c}1\\ 0\end{array}\right).$$

$$\text{H}_{1}=\left\{\left(\begin{array}[]{cc}1&0\\ 0&e^{i\beta}\end{array}\right),0\leq\beta<2\pi\right\}\subset\text{U(2)}.$$

$$\psi_{2}=e^{i\varphi_{12}}\left(\begin{array}[]{c}C_{12}\\ S_{12}\end{array}\right),$$

$$\psi_{3}=e^{-i\varphi_{31}}\left(\begin{array}[]{c}C_{31}\\ e^{i\phi}S_{31}\end{array}\right),\quad 0\leq\phi<2\pi,$$

$$({\psi }_2,{\psi }_3)=e^{i{\phi }_{23}}{C}_{23}=e^{-i({\phi }_{12}+{\phi }_{31})}(C_{12}{C}_{31}+e^{i\varphi }{S}_{12}{S}_{31}),$$

$$i.e.,C_{23}=e^{i{\phi }_g}(C_{12}{C}_{31}+e^{i\varphi }{S}_{12}{S}_{31}).$$

$$|C_{12}{C}_{31}+e^{i\varphi }{S}_{12}{S}_{31}|\le C_{12}{C}_{31}+S_{12}{S}_{31}=\cos \frac{1}{2}({\theta }_{12}-{\theta }_{31})\le 1,$$

$${\phi }_{23}=-({\phi }_g+{\phi }_{12}+{\phi }_{31}).$$

$$[\hat{q},\hat{p}]=i.1;$$

$$\hat{a}=\frac{1}{\sqrt{2}}(\hat{q}+i\hat{p}),{\hat{a}}^{†}=\frac{1}{\sqrt{2}}(\hat{q}-i\hat{p}):[\hat{a},{\hat{a}}^{†}]=1.$$

$$n=0,1,2,\dots :|n⟩=\frac{({\hat{a}}^{†}{)}^n}{\sqrt{n!}}|0⟩,{\hat{a}}^{†}\hat{a}|n⟩=n|n⟩;$$

$$z\in \mathbb{C}:|z⟩=e^{-\frac{1}{2}|z{|}^2}\sum _{n=0}^{\infty }\frac{z^n}{\sqrt{n}!}|n⟩,\hat{a}|z⟩=z|z⟩.$$

$$⟨n^{\prime }|n⟩={\delta }_{n^{\prime },n},\sum _{n=0}^{\infty }|n⟩⟨n|=1;$$

$$⟨z^{\prime }|z⟩=e^{-\frac{1}{2}|z^{\prime }-z{|}^2+i\mathrm{Im}(z^{\prime \ast }z)},\int \frac{d^{2}z}{\pi }|z⟩⟨z|=1.$$

$$\mathcal{M}=\{e^{i\alpha }|z⟩|0\le \alpha <2\pi ,z\in \mathbb{C}\}.$$

$$U({\alpha }_0,z_0,{\theta }_0)=e^{i{\alpha }_0}D(z_0)e^{-i{\theta }_0{\hat{a}}^{†}\hat{a}}\in {\text{G}}_4,D(z_0)=e^{z_0{\hat{a}}^{†}-z_0^{\ast }\hat{a}}:$$

Full text

The geometric phase was originally discovered in a quantum mechanical context based on several physically reasonable assumptions 1 . Thereafter our understanding of it has evolved in several steps to a level where many of the original assumptions have been shown to be unnecessary 2 . In the kinematic approach the minimal conditions needed to be able to define the geometric phase have been identified 3 . In the process the role of the Bargmann Invariants, and later of the so-called Null Phase Curves as basic ingredients of the theory, have been clarified 4 . The original connection between Bargmann Invariants and geometric phases based on geodesics in quantum mechanical ray spaces has been greatly enlarged by showing that the geodesics can be replaced by the much more numerous Null Phase Curves.

Bargmann Invariants (BI) are of various integral orders, the lowest nontrivial one being of order three. The fourth and higher order BI’s can in principle be reduced to those of third order, which is thus the primitive one. It is therefore natural to study these in some detail. It has recently been pointed out that for finite dimensional systems the general properties of these BI’s are not much known, especially for high dimensions 5 . It has then been shown that the calculation of these BI’s can be handled in a uniform and efficient manner using Majorana’s theorem concerning symmetric SU(2) multispinors 6 . The calculations go back to computing solid angles on Poincaré spheres familiar from polarization optics.

The purpose of the present work is to explore the properties and parametrisations of the third order Bargmann Invariant. We show how its intrinsic invariance and other algebraic properties can be brought out elegantly using group theoretical methods. In particular it can be parametrized in a natural and intrinsic manner using unitary invariant angle variables. We bring out the dependence of the algebraic properties of these variables on the state space dimension. We then combine the Majorana theorem with the Schwinger oscillator construction of SU(2) representations 7 , and study both Bargmann Invariants and Null Phase Curves in this new framework.

The contents of this paper are organised as follows. Section I recalls definitions of BI’s and geometric phases, and the connection between them based on ray space geodesics. It also introduces a natural set of six angle parameters associated with any third order BI, which are invariant under all unitary transformations. Section II examines the extent to which these angles are algebraically independent. Using group theoretic methods, it is shown that while for two-dimensional systems only five of the six angles are independent, for dimensions three and higher all six are independent. Interestingly it is shown that for the subset of coherent states of a one-dimensional oscillator, only five of the six angles are independent. Section III describes briefly the family of Null Phase Curves (NPC) in Hilbert and ray spaces which have been shown to be basic to geometric phase theory. The definition, important properties and procedure for construction of NPC’s are given in a concise manner. Section IV studies both BI’s and NPC’s for finite dimensional systems using the Schwinger–Majorana framework. This is a combination of the Schwinger oscillator treatment of quantum angular momentum theory, and the Majorana theorem on symmetric SU(2) multispinors. Section V contains some Concluding Remarks, and the Appendix presents the basic features of the Schwinger–Majorana framework which allows a uniform description of all finite dimensional Hilbert spaces.

I Three-vertex Bargmann Invariants – invariances, intrinsic parameters, connection to geometric phases

Let $\mathcal{H}$ be a complex Hilbert space, of finite or infinite dimension, pertaining to some quantum system. Vectors in $\mathcal{H}$ are $\psi,\phi,\ldots$, and the inner product is $(\phi,\psi)$ or $\langle\phi|\psi\rangle$. The unit sphere $\mathcal{B}\subset\mathcal{H}$, and the space $\mathcal{R}$ of unit rays, are

[TABLE]

Neither $\mathcal{B}$ nor $\mathcal{R}$ is a linear space, they are related by a projection map $\pi$:

[TABLE]

Thus $\mathcal{B}$ is a U(1) bundle over $\mathcal{R}$ as base.

If the complex dimension of $\mathcal{H}$ is finite, $n$ say, then $\mathcal{B}$ and $\mathcal{R}$ are spaces of real dimensions $(2n-1),2(n-1)$ respectively.

Let $\psi_{j},j=1,2,3$ be any three vectors in $\mathcal{B}$, pairwise linearly independent and nonorthogonal. They define the third order or three-vertex Bargmann invariant (BI)

[TABLE]

which is basic to geometric phase theory. This expression is nonzero, in general complex, and is actually defined on $\mathcal{R}$ since

[TABLE]

The relation to geometric phases, as originally established, arises as follows 8 . For any two ‘nonorthogonal’ points $\rho(\psi_{1}),\rho(\psi_{2})$ in $\mathcal{R}$, there is a (unique shorter) geodesic (with respect to the Fubini-Study metric on $\mathcal{R}$) connecting them. Choose $\psi_{1},\psi_{2}$ to be ‘in phase’ in the Pancharatnam sense 9 :

[TABLE]

Then the geodesic $C_{{\rm geo},12}$ from $\rho(\psi_{1})$ to $\rho(\psi_{2})$ is the projection by $\pi$ of a parametrised curve $\mathcal{C}_{{\rm geo},12}$ in $\mathcal{B}$ from $\psi_{1}$ to $\psi_{2}$:

[TABLE]

Now given the vertices $\rho(\psi_{1}),\rho(\psi_{2}),\rho(\psi_{3})$ of $\Delta_{3}(\psi_{1},\psi_{2},\psi_{3})$, connect them by successive geodesics $C_{{\rm geo},12},C_{{\rm geo},23}$ and $C_{{\rm geo},31}$. (Of course we cannot choose $\psi_{1},\psi_{2},\psi_{3}$ to all obey conditions like $(\ref{2.5})$ in general!) Then their union

[TABLE]

gives a geodesic triangle in $\mathcal{R}$, a closed cyclic evolution in the quantum mechanical state space. This evolution can be produced by a suitable (time dependent) Hamiltonian operator via the Schrödinger equation, and the corresponding geometric phase turns out to be

[TABLE]

Actually, the use of a Hamiltonian and Schrödinger equation are not essential; this connection between geometric phases and BI’s can be understood more directly in the kinematic approach. We return to this and to its nontrivial generalisation in the sequel.

At this point we turn to a study of the invariances and intrinsic properties of the BI $\Delta_{3}(\psi_{1},\psi_{2},\psi_{3})$. If $U$ is any unitary transformation on $\mathcal{H}$, we have the obvious invariance property:

[TABLE]

The BI itself can be parametrised by six intrinsic angle type variables, two for each factor:

[TABLE]

The six variables $\theta_{jk},\varphi_{jk}$ are all defined at the level of $\mathcal{B}$, and are of course invariant under the transformations $(\ref{2.9})$, i.e., they are unitary invariants. They are intrinsic to the triad $\{\psi_{j}\}$. If we make independent phase changes in the $\psi_{j}$, the $\theta_{jk}$ are unchanged while the $\varphi_{jk}$ change in a simple way:

[TABLE]

Thus the individual $\varphi_{jk}$ are not ray space quantities but their sum is invariant under $(\ref{2.11})$, and according to $(\ref{2.8})$ is a geometric phase:

[TABLE]

We can see that the six angles $\theta_{jk},\varphi_{jk}$ determine the triad $\{\psi_{j}\}$, equivalently their configuration, upto an overall unitary transformation $U$ on $\mathcal{H}$. The important question is the extent to which they are algebraically independent.

II Algebraic independence of Bargmann Invariant parameters in two and three dimensions, the case of coherent states

The question we wish to answer is this: can the values of $\theta_{jk},\varphi_{jk}$ be chosen independently, and will they then lead to a definite triad of vectors $\{\psi_{j}\}$ upto an overall unitary transformation? We will find that the answer depends on the dimension of $\mathcal{H}$, and also on whether any overall restrictions are placed on the possible choices of the $\psi_{j}$.

The case $\dim\mathcal{H}=n=2$

The full unitary group on $\mathcal{H}$ is the four-parameter U(2), and the $\psi_{j}^{\prime}s$ can be represented by two component complex column vectors in an unspecified orthonormal basis (ONB). In any case, $\psi_{3}$, say, can be expressed as a linear combination of $\psi_{1}$ and $\psi_{2}$ as the latter are linearly independent, with coefficients determined by $\theta_{jk},\varphi_{jk}$:

[TABLE]

Such a relation need not hold in higher dimensions.

To proceed further we use a group theoretic approach. Using the freedom of U(2) action we can assume without loss of generality that in some ONB

[TABLE]

The stability group of this $\psi_{1}$ is a one-dimensional U(1) subgroup $\text{H}_{1}\subset$ U(2):

[TABLE]

For independently given angles $\theta_{12},\varphi_{12}$ we can use $\text{H}_{1}$ action to achieve

[TABLE]

after which there is no more U(2) action possible. Then for independently given angles $\theta_{31},\varphi_{31}$ the vector $\psi_{3}$ is necessarily of the form

[TABLE]

bringing in one additional angle $\phi$ independent of $\theta_{12},\theta_{31},\varphi_{12},\varphi_{31}$. Thus a given triad $\{\psi_{j}\}$ leads to exactly five independent U(2) invariant angles $\theta_{12},\theta_{31},\varphi_{12},\varphi_{31},\phi$. The two remaining U(2) invariant angles $\theta_{23},\varphi_{23}$ are to be found from

[TABLE]

Since

[TABLE]

we see that $\theta_{23}$ and $\varphi_{g}$ are unambiguously determined by eq. $(\ref{3.6})$ in terms of $\theta_{12},\theta_{31},\phi$ and so cannot be independently chosen. Once $\varphi_{g}$ has been determined, $\varphi_{23}$ follows from

[TABLE]

We thus find that five algebraically independent intrinsic U(2) invariant angles can be chosen in several equivalent ways: $\theta_{12},\theta_{31},\phi,\varphi_{12},\varphi_{31}$ or $\theta_{12},\theta_{31},\theta_{23},\varphi_{12},\varphi_{31}$ or $\theta_{12},\theta_{31},\varphi_{23},\varphi_{12},\varphi_{31}$. (Other choices are also easily found). In any case, the six U(2) invariant angles $\theta_{jk},\varphi_{jk}$ are not algebraically independent. Each of the three choices of algebraically independent angles listed above is of course at the level of the space $\mathcal{B}$. Taking for example the second set consisting of $\theta_{12},\theta_{31},\theta_{23},\varphi_{12},\varphi_{31}$, we see that while the $\theta_{jk}$ ‘descend’ to the ray space $\mathcal{R}$, neither $\varphi_{12}$ nor $\varphi_{31}$ do so since under independent phase changes $(\ref{2.11})$ in the vectors $\psi_{j}$ they are not preserved but change independently. Thus a triad of vertices $\rho(\psi_{j}),j=1,2,3$ in $\mathcal{R}$, determining a geodesic triangle in $\mathcal{R}$, is determined intrinsically by exactly three algebraically independent angles $\theta_{jk}$. For dim $\mathcal{H}=2,\mathcal{R}$ is the (Poincaré) sphere $S^{2}$, and as is well known a geodesic triangle on $S^{2}$ is intrinsically determined by three independent angle parameters. It was for the cyclic evolution of the polarization state of plane electromagnetic waves around such a triangle on $S^{2}$ that the geometric phase was found in Pancharatnam’s pioneering work 9 .

The case of one-dimensional coherent states

It is interesting that a remarkably similar situation occurs in the case of the infinite dimensional Hilbert space $\mathcal{H}=L^{2}(\mathbb{R})$, in the context of the coherent states. This Hilbert space carries an irreducible representation of the Heisenberg canonical commutation relation for hermitian operators $\hat{q},\hat{p}$ or their nonhermitian combinations $\hat{a},\hat{a}^{\dagger}$:

[TABLE]

The eigenstates of $\hat{a}^{\dagger}\hat{a}$ (Fock states), and the right eigenstates of $\hat{a}$ (coherent states), are related to one another as follows:

[TABLE]

The former give an ONB in $\mathcal{H}$, the latter are nonorthogonal and overcomplete:

[TABLE]

Now define a subset $\mathcal{M}\subset\mathcal{H}$, not a subspace, by

[TABLE]

There is a natural four parameter group $\text{G}_{4}$ of unitary transformations on $\mathcal{H}$ which is generated by $\{\mathbb{I},\hat{q},\hat{p},\hat{a}^{\dagger}\hat{a}\}$ and acts transitively on $\mathcal{M}$:

[TABLE]

(The composition law in $\text{G}_{4}$ is easily found but not needed for our purposes).

Let now $|\psi_{j}\rangle,j=1,2,3$ be a triad of vectors with the restriction that each of them belong to $\mathcal{M}$, and let the six angles $\theta_{jk},\varphi_{jk}$ be as in eq. $(\ref{2.10})$. These angles are of course invariant under $\text{G}_{4}$ action. Using a suitable $\text{G}_{4}$ element we can begin by assuming without loss of generality

[TABLE]

The stability group of this $|\psi_{1}\rangle$ is the U(1) subgroup

[TABLE]

Now for any given independent $\theta_{12},\varphi_{12}$ we can use $\text{H}_{1}$ action to achieve

[TABLE]

At this point, all of $\text{G}_{4}$ has been ‘used up’. Now given any $\theta_{31},\varphi_{31}$ independent among themselves and of $\theta_{12},\varphi_{12}$, we find that $|\psi_{3}\rangle$ necessarily has the form

[TABLE]

Thus one additional angle $\phi^{\prime}$ independent of $\theta_{12},\theta_{31},\varphi_{12},\varphi_{31}$ has appeared (compare with eq. $(\ref{3.5}))$, so any triad $\{|\psi_{j}\rangle\}$ of the prescribed type is characterised by exactly five independent $\text{G}_{4}$ invariant angles $\theta_{12},\theta_{31},\varphi_{12},\varphi_{31},\phi^{\prime}$. The remaining two $\text{G}_{4}$ invariant angles $\theta_{23},\varphi_{23}$ are to be found from

[TABLE]

Since

[TABLE]

we see that $\theta_{23}$ and $\varphi_{g}$ are unambiguously determined by eq. $(\ref{3.18})$ in terms of $\theta_{12},\theta_{31},\phi^{\prime}$. Once $\varphi_{g}$ has been determined, $\varphi_{23}$ follows from eq. $(\ref{3.8})$ again.

In this example, then, out of the six angles $\theta_{jk},\varphi_{jk}$ only five are algebraically independent. As independent sets we can choose for example $\theta_{12},\theta_{31},\phi^{\prime},\varphi_{12},\varphi_{31}$ or $\theta_{12},\theta_{31},\theta_{23},\varphi_{12},\varphi_{31}$, etc. The overall similarity to the case of $\dim\mathcal{H}=2$ is perhaps unexpected.

At this point we return to $\mathcal{H}$ of finite dimensions.

The case $\dim\mathcal{H}=n=3$

Now we find that all six angles $\theta_{jk},\varphi_{jk}$ are algebraically independent. Unitary transformations on $\mathcal{H}$ constitute the nine-parameter group U(3); using its action we can arrange

[TABLE]

in a suitable ONB. The stability group of $\psi_{1}$ is a U(2) subgroup in U(3):

[TABLE]

For given independent choices of $\theta_{12},\varphi_{12}$ we use $\text{H}_{1}$ action to achieve

[TABLE]

The stability group for the pair $\psi_{1},\psi_{2}$ is a U(1) subgroup:

[TABLE]

Now for given independent choices of $\theta_{31},\varphi_{31}$ we use $\text{H}_{2}$ action to achieve

[TABLE]

This brings in two additional angles $\phi,\xi$ independent of $\theta_{12},\theta_{31},\varphi_{12},\varphi_{31}$; and all the freedom of U(3) action has been ‘used up’. The triad $\{\psi_{j}\}$ is intrinsically characterised by six independent U(3) invariant angles $\theta_{12},\theta_{31},\varphi_{12},\varphi_{31},\phi,\xi$. The two remaining original U(3) invariant angles $\theta_{23},\varphi_{23}$ are to be found from

[TABLE]

In contrast to eq. $(\ref{3.6})$, now (for any given $\theta_{12},\theta_{31}$) the pair $\theta_{23},\varphi_{g}$ is unambiguously determined by the pair $\phi,\xi$ and so they can be independently specified. We are free to choose $\theta_{12},\theta_{31},\varphi_{12},\varphi_{31},\phi,\xi$ or $\theta_{jk},\varphi_{jk}$ as six independent U(3) invariants characterising $\{\psi_{j}\}$.

As long as we deal with the three-vertex BI $\Delta_{3}(\psi_{1},\psi_{2},\psi_{3})$, the situation for $\dim\mathcal{H}=n\geq 4$ is the same as for $n=3$, since $\psi_{1},\psi_{2},\psi_{3}$ always lie in some three-dimensional subspace in $\mathcal{H}$.

Geometric phase formulae for geodesic triangles

In Hilbert spaces of dimensions 2 and 3 the geometric phase results of eqs. $(\ref{2.8},\ref{2.12})$ can be given more explicitly. For the passages to the ray spaces, $\mathcal{B}\rightarrow\mathcal{R}$, in the two cases, we use eqs. $(\ref{3.2},\ref{3.4},\ref{3.5})$ and eqs. $(\ref{3.20},\ref{3.22},\ref{3.24})$ respectively, drop overall phase factors, and obtain expressions for the vertices:

[TABLE]

For $n=2$, a general geodesic triangle on $\mathcal{R}=\mathbb{S}^{2}$ is intrinsically characterized by three independent angle parameters $\theta_{12},\theta_{31},\phi$. This is familiar from spherical trigonometry on $\mathbb{S}^{2}$. For $n=3$, ray space $\mathcal{R}$ is a simply connected four dimensional region $\vartheta\subset\mathbb{S}^{7}\subset\mathbb{R}^{8}$, a (small) portion of the unit sphere in real eight dimensional Euclidean space 10 . A geodesic triangle on $\vartheta$ is intrinsically characterized by four independent angle parameters $\theta_{12},\theta_{31},\phi,\xi$. From eqs. $(\ref{3.6},\ref{3.25})$ the geometric phases are:

[TABLE]

For $n=2$, this is the original result of Pancharatnam; for $n=3$ we have a genuine generalisation of Pancharatnam’s result 9 , 10 .

III Definition, properties, construction of Null Phase Curves, generalized Bargmann Invariant-geometric phase connection

It was mentioned in Section I that an extensive generalisation of the BI-geometric phase connection $(\ref{2.8})$ exists. It turns out that the ray space geodesics appearing on the left hand side of eq. $(\ref{2.8})$ can each be replaced by a so-called Null Phase Curve (NPC), with no change on the right hand side. Thus, the (negative of the) phase of a BI is the geometric phase for many cyclic ray space evolutions, not just for the one along the sides of a geodesic triangle.

It is an interesting fact that for $\dim\mathcal{H}=2$, NPC’s coincide with geodesics. However, for $\dim\mathcal{H}\geq 3$, given any two nonorthogonal points $\rho_{1},\rho_{2}\in\mathcal{R}$, there exist infinitely many NPC’s connecting them, the geodesic being just one of them. In this Section we define, describe and outline the construction of the most general NPC connecting any two given non-orthogonal points in ray space. This is a summary of the results in references 4 . The spaces $\mathcal{B},\mathcal{R}$ associated with $\mathcal{H}$ will be used extensively.

The notations for continuous parametrized curves in $\mathcal{B}$ and $\mathcal{R}$ are as follows:

[TABLE]

Any $\mathcal{C}$ projecting onto a given $C$ is a lift of the latter. The prerequisites for the geometric phase to be defined for $C,\mathcal{C}$ are:

[TABLE]

(Here $C$ need not be closed; and even if it is, $\mathcal{C}$ could be open). The prerequisites for $C$ to be a NPC are:

[TABLE]

For such $C$ we will use lifts $\mathcal{C}$ obeying similar conditions:

[TABLE]

Definition of a NPC

Given conditions (4.3, 4.4) we have two equivalent ways to define a NPC. $C\subset\mathcal{R}$ is a NPC if either

[TABLE]

or if for any fixed $s_{0}\in[s_{1},s_{2}]$,

[TABLE]

We denote NPC’s in $\mathcal{R}$ by $N,N^{\prime},\ldots$. Any lift of $N$ will be written as $\mathcal{N}$, and will also be called a NPC (in $\mathcal{B}$).

III.1 Properties of NPC’s

For any lift $\mathcal{N}$ of a NPC $N\subset\mathcal{R}$ from $\rho_{1}$ to $\rho_{2}$,

[TABLE]

For any NPC $N\subset\mathcal{R}$, there exist lifts $\mathcal{N}_{0}\subset\mathcal{B}$ such that

[TABLE]

Construction of most general NPC

Let $\rho_{1},\rho_{2}$ be distinct nonorthogonal points in $\mathcal{R}$. Choose $\psi_{1},\psi_{2}\in\mathcal{B}$ projecting on to them and obeying the Pancharatnam ‘in phase’ condition $(\ref{2.5})$ which is repeated here:

[TABLE]

Then there exists an ON pair of vectors $\{e_{1},e_{2}\}$ such that

[TABLE]

The (unique) geodesic from $\psi_{1}$ to $\psi_{2}$ is a reparametrised form of eq. $(\ref{2.6})$:

[TABLE]

The steps to follow to obtain a lift $\mathcal{N}_{0}$ (of the type obeying eq. $(\ref{4.8})$) of an NPC $N$ from $\psi_{1}$ to $\psi_{2}$ are these:

(a) Extend the ON pair $\{e_{1},e_{2}\}$ to an ONB $\{e_{1},e_{2},e_{3},\ldots,e_{n}\}$ for $\mathcal{H}$ in any way.

(b) Define $\psi_{0}(s),s\in[s_{1},s_{2}]$, by

[TABLE]

where

[TABLE]

is a real $n$-component vector obeying three types of conditions:

(c) Boundary conditions:

[TABLE]

(d) Local conditions: for $s\in(s_{1},s_{2})$,

[TABLE]

Thus for all $s\in[s_{1},s_{2}],{\bf x}(s)\in\mathbb{S}^{n-1}$.

(e) Nonlocal conditions:

[TABLE]

Then

[TABLE]

is a lift (of the type $(\ref{4.8})$) of an NPC $N\subset\mathcal{R}$ from $\rho_{1}$ to $\rho_{2}$; and all possible $N$ will be obtained in this way.

The most general solution to the nonlocal condition $(\ref{4.16})$ cannot be easily given explicitly in any way. If we restrict the choice of ${\bf x}(s)$ by the condition $x_{r}(s)\geq 0,r=2,3,\ldots,n$, conditions $(\ref{4.16})$ are immediately obeyed. However, in the most general case, $x_{r}(s)$ for some $r\in(2,3,\ldots n)$ can be negative for some ranges of $s$.

Step (a) in the above construction shows why nongeodesic NPC’s exist only for $n\geq 3$.

Generalised BI-geometric phase connection

This is an extension of eq. $(\ref{2.8})$. We can replace each of the geodesics on the left by any NPC with the same endpoints. Use of the property $(\ref{4.7})$ of (lifts of) NPC’s leads in an obvious notation to the result

[TABLE]

IV Bargmann invariants and Null Phase Curves in the Schwinger-Majorana SU(2) framework

The widest possible generalisation of the original BI-geometric phase connection $(\ref{2.8})$ is given by eq. $(\ref{4.18})$. On the left hand side, geodesics have been replaced by the much more plentiful NPC’s (for each given pair of vertices). On the right, the BI stays unchanged. Here it is of course assumed that $\dim\mathcal{H}=n\geq 3$.

In this Section we apply the ideas of the Schwinger-Majorana framework for SU(2) to both sides of the connection $(\ref{4.18})$. We first deal with the right hand side for general $n$, and see how it reduces in principle to calculations on Poincaré spheres. The case $n=3$ is worked out in some detail. Next we study the left hand side, for $n=3$, to see how the difference between geodesics and general NPC’s appears in the Schwinger-Majorana framework.

Treatment of the BI

In dealing with vectors in $\mathcal{H}_{n}^{(Sch)}$, we have the freedom to use unitary transformations within the UIR $D^{(J)}(u)$ of SU(2), or to use the wider set of transformations in U(n). (In what follows, $n$ is kept fixed). To study the BI $\Delta_{3}(\psi_{1},\psi_{2},\psi_{3})$ in general terms, in the spirit of Section II we exploit U(n) action. However, at the same time we use the information about the description of vectors from the SU(2) point of view, given in the Appendix.

Let the triad $|\psi_{j}\rangle\in\mathcal{H}_{n}^{(Sch)},j=1,2,3$, determine the six intrinsic and algebraically independent angles $\theta_{jk},\varphi_{jk}$ as in eq. $(\ref{2.10})$. In the present situation, a preferred ONB $(\ref{A.4})$ in $\mathcal{H}_{n}^{(Sch)}$ is already given by SU(2) considerations. With the help of a suitable U(n) transformation we can always map $|\psi_{1}\rangle$ to the highest weight vector $|J,J\rangle$. We indicate this by

[TABLE]

To preserve the BI this same U(n) transformation must be applied to $|\psi_{2}\rangle,|\psi_{3}\rangle$ as well. To avoid making the notation excessively intricate, such steps will be left implicit.

The stability group of $|\psi^{\prime}_{1}\rangle$ is clearly a U(n-1) subgroup of U(n). We can next always use this U(n-1) action (after the U(n) action $(\ref{5.1})$) to map $|\psi_{2}\rangle$ to some pure product state $|\xi;n\rangle$:

[TABLE]

provided (see eq. $(\ref{A.15})$)

[TABLE]

This essentially determines $\alpha$ but leaves the phase of $\beta$ undetermined:

[TABLE]

Turning finally to $|\psi_{3}\rangle$, it is clear that in general we cannot expect to be able to map it to some pure product state $|\xi^{\prime};n\rangle$, because (as shown in Section II) with two-dimensional systems only five algebraically independent angles can be accommodated. Thus the best that can be achieved in general, using U(n) action, is that $|\psi_{1}\rangle,|\psi_{2}\rangle$ go into pure product states while $|\psi_{3}\rangle$ goes into a general Majorana state:

[TABLE]

where $c$ is a normalising constant:

[TABLE]

Then, given $\theta_{31}$ and $\varphi_{31}$, use of eq. $(\ref{A.16})$ leads to

[TABLE]

giving partial information on $\{\xi^{\prime}\}$. The phase freedom of $\beta$ in eq. $(\ref{5.4})$, and in the choice of $\{\xi^{\prime}\}$ after requiring eq. $(\ref{5.7})$, are analogues of the presence of independent angles $\phi$ in eq. $(\ref{3.5})$, $\phi$ and $\xi$ in eq. $(\ref{3.24})$. All these remaining freedoms while reproducing given $\theta_{jk},\varphi_{jk}$ must be kept in mind.

Now we turn to the BI which by U(n) invariance becomes

[TABLE]

The righthand side of eq. $(\ref{4.18})$ then becomes

[TABLE]

so the general $n$-level system geometric phase as viewed from the BI is the sum of $(n-1)$ geometric phases of Pancharatnam type computed using $(n-1)$ geodesic triangles on the Poincaré’ sphere 11 .

This result is of a mathematical nature, without implying any physical substructure for the $n$-level quantum system. It may also be viewed as a general structure analysis of the $n$-level three-vertex geometric phase, not as an explicit evaluation of it in the sense of eqs. $(\ref{3.27})$.

It is instructive to work out the above expressions in more detail for the lowest nontrivial case $n=3$. The spin 1 UIR of SU(2), $D^{(1)}(u)$, as well as the nine parameter group U(3), act on $\mathcal{H}_{3}^{(Sch)}$. (The former is equivalent to the real defining representation of SO(3)). From the SU(2) point of view, we have the following ONB and vector descriptions:

[TABLE]

Out of the ONB vectors, only $e_{1}$ and $e_{3}$ are of pure product type. The general vector $|\{\xi,\xi^{\prime}\},3\rangle$ determines an unordered pair of points $\{\hat{n},\hat{n}^{\prime}\}$ on the Poincaré sphere:

[TABLE]

For the pure product vector, $\hat{n}^{\prime}=\hat{n}$. So for the ONB vectors we have

[TABLE]

Now let $\psi_{j},j=1,2,3$ be a triad of (normalized) vectors in $\mathcal{H}_{3}^{(Sch)}$, with the set of six independent U(3) invariant angles $\theta_{jk},\varphi_{jk}$. From eq. $(\ref{5.1})$ we know that with no loss of generality we can assume

[TABLE]

From eqs $(\ref{5.2},\ref{5.3},\ref{5.4})$ we may next assume $\psi_{2}$ is also of pure product type:

[TABLE]

Here we have chosen $\arg\beta_{2}$ conveniently, and then $\theta_{12},\varphi_{12}$ are properly incorporated. The choice of a convenient form for $\psi_{3}$ is more involved. From eq. ($\ref{5.5})$ we know that in general it is of general type:

[TABLE]

Now on the one hand $\xi_{3},\xi^{\prime}_{3}$ are limited by the given values of $\theta_{23},\theta_{31},\varphi_{23},\varphi_{31}$:

[TABLE]

These conditions determine $\psi_{3}$, i.e. $\{\xi_{3},\xi^{\prime}_{3}\}$, upto a one-parameter group of U(1) transformations in the one-dimensional subspace of $\mathcal{H}_{3}^{(Sch)}$ orthogonal to both $\psi_{1}$ and $\psi_{2}$. However, for all these $\psi_{3}$’s the geometric phase is the same, as seen explicitly in eqs. $(\ref{3.1})$. On the other hand they determine the pair of points on the Poincaré sphere corresponding to $\psi_{3}$:

[TABLE]

From eqs $(\ref{5.13},\ref{5.14})$, both $\hat{n}_{1}=\hat{n}^{\prime}_{1}$ and $\hat{n}_{2}=\hat{n}^{\prime}_{2}$ lie on the ‘Greenwich’ meridian with azimuthal angle $\phi=0$. To then allow for the most general choice of $\psi_{3}$ we must allow $\hat{n}_{3},\hat{n}^{\prime}_{3}$ to be any independently chosen pair of points on the Poincaré sphere. Using spherical polar variables we thus have

[TABLE]

When the two complex equations $(\ref{5.16})$ are viewed as conditions on $\hat{n}_{3},\hat{n}^{\prime}_{3}$ the earlier remarks tell us that the four real angles $\theta_{3},\theta^{\prime}_{3},\phi_{3},\phi^{\prime}_{3}$ are determined by $\theta_{23},\theta_{31},\varphi_{23},\varphi_{31}$ upto the freedom of U(1) transformations mentioned above, but this does not affect the geometric phase. For the calculation of this phase, and picturing it on the Poincaré sphere, the former angles are more convenient. From Pancharatnam’s well known result for two-level systems, eq. $(\ref{5.9})$ becomes

[TABLE]

where each $\Omega(\ldots)$ is the solid angle of the spherical triangle with indicated vertices (counted positive if the sequence of vertices appears anticlockwise when viewed from the outside).

NPC’s in Schwinger–Majorana framework – general structure analysis

To keep the various expressions as simple as possible, we consider only the case of $\mathcal{H}_{3}^{(Sch)}$ corresponding to spin $J=1$. This is the lowest dimension in which nontrivial NPC’s occur.

Given two normalised vectors $\psi_{1},\psi_{2}\in\mathcal{H}_{3}^{(Sch)}$ with $(\psi_{1},\psi_{2})=\cos\frac{1}{2}\theta_{0},\theta_{0}\in(0,\pi)$, we wish to describe a general NPC, and contrast it with the geodesic, connecting them. (The latter is given, in the case of any Hilbert space $\mathcal{H}$, by eq. $(\ref{2.6})$.) In each case at each point we wish to find and visualise the corresponding unordered pair $\{\hat{n}^{\prime},\hat{n}\}$ of points on $\mathbb{S}^{2}$. All this will use the review of NPC’s in Section III.

From the pair $\psi_{1},\psi_{2}$ we extract an ON pair written as $\{e^{\prime}_{1},e^{\prime}_{2}\}$ (to be distinguished from $e_{1},e_{2}$ already used in eq. $(\ref{5.10}(a))$), and have in place of eq. $(\ref{4.10})$

[TABLE]

From the U(3) point of view there is no intrinsic difference or distinction between one ON pair and another, as one can be transformed into the other. However from the SU(2) point of view there are intrinsically distinct possibilities which cannot be connected by SU(2) transformations. (As can easily be verified, SU(2) transformations on $\mathcal{H}_{3}^{(Sch)}$ take pure product states into other pure product states, and general (Majorana) vectors to other such vectors.) Thus each of $e^{\prime}_{1}$ and $e^{\prime}_{2}$ can be a pure product vector or a general (Majorana) vector. We now analyse two of these three cases, to illustrate the kinds of configurations that can arise.

The pure–pure case

Suppose both $e^{\prime}_{1}$ and $e^{\prime}_{2}$ are pure product states:

[TABLE]

Then both belong to the SU(2) orbit of $e_{1}$. We can easily see that by a suitable SU(2) transformation followed by another suitable U(3) transformation, this pair can be mapped to the pair $e_{1},e_{3}$ of eq. $(\ref{5.10}$(a)):

[TABLE]

Let us for convenience use the short hand symbols

[TABLE]

all of which are nonnegative for $0\leq s\leq 1$. Then we assume

[TABLE]

The geodesic $\mathcal{C}_{geo}$ connecting them is

[TABLE]

The Majorana pair of vectors $\{\hat{n}^{\prime}(s),\hat{n}(s)\}$ on $\mathbb{S}^{2}$ representing $\psi(s)$ along $\mathcal{C}_{geo}$ is then

[TABLE]

Both vectors are on the 2–3 meridian of $\mathbb{S}^{2}$, reflections of each other in the 1–3 plane.

The most general NPC $\mathcal{N}_{0}$ from $\psi_{1}$ to $\psi_{2}$, obeying eq. $(\ref{4.8})$, is constructed following the sequence of steps in Section III, eqs. $(\ref{4.12}-\ref{4.17})$. The first step is to extend $\{e_{1},e_{3}\}$ to an ONB for $\mathcal{H}_{3}^{(Sch)}$ in the most general way. Thus for any fixed $\eta$, we adjoin $e^{i\eta}e_{2}$ to $\{e_{1},e_{3}\}$. Then the most general $\mathcal{N}_{0}$ is

[TABLE]

where ${\bf x}(s)$ obeys:

[TABLE]

in addition to being continuous once-differentiable. Compared to the geodesic $(\ref{5.25})$ where ${\bf x}(s)=(C,S,0)$, it is $x_{3}(s)$ that is new. For the Majorana pair along $\mathcal{N}_{0}$ we need to factorize the quadratic in $\psi_{0}(s)$:

[TABLE]

Therefore we have the pair (upto real normalisation factors)

[TABLE]

leading to corresponding $\hat{n}^{\prime}_{0}(s),\hat{n}_{0}(s)$ on $\mathbb{S}^{2}$. These expressions are somewhat complicated, but compared to eqs $(\ref{5.25},\ref{5.26})$ for the geodesic some differences show up: the points $\hat{n}^{\prime}_{0}(s),\hat{n}_{0}(s)$ are generally not on the 2–3 meridian, and not reflections of one another in the 1–3 plane, but have general positions on $\mathbb{S}^{2}$.

The pure–general case

This occurs when in eq. $(\ref{5.20})$, $e^{\prime}_{1}$ (say) is a pure product while $e^{\prime}_{2}$ is not:

[TABLE]

Now one can see that by a suitable SU(2) transformation $e^{\prime}_{1}$ can be transformed to $e_{1}$, and therefore at the same time $e^{\prime}_{2}$ goes into some normalised linear combination of $e_{2}$ and $e_{3}$. At the next step by a suitable U(3) transformation preserving $e_{1},e^{\prime}_{2}$ can be taken to $e_{2}$. In this way in place of eq. $(\ref{5.22})$ we achieve

[TABLE]

and in place of eq. $(\ref{5.24})$ we have the pair

[TABLE]

The geodesic $\mathcal{C}_{geo}$ connecting them is

[TABLE]

In contrast to eq. $(\ref{5.25})$ in the pure-pure case, both of these are real. The Majorana pair $\{\hat{n}^{\prime}(s),\hat{n}(s)\}$ is therefore

[TABLE]

Here again the contrast with eq. $(\ref{5.26})$ is evident.

For the general NPC from $\psi_{1}$ to $\psi_{2}$ in this case we follow steps similar to the previous pure-pure case. The replacements for eqs. $(\ref{5.27}-\ref{5.30})$ are:

[TABLE]

The conditions on ${\bf x}(s)$ are identical to those given in eq. $(\ref{5.28})$. Again, compared to the geodesic $(\ref{5.34})$ where ${\bf x}(s)=(C,S,0),x_{3}$ is new. For the Majorana pair we find:

[TABLE]

leading to the pair (upto normalisation)

[TABLE]

This is to be compared on the one hand to the geodesic pair $(\ref{5.34})$, and on the other hand to the pure-pure NPC case $(\ref{5.30})$. In the pair $\{\hat{n}^{\prime}_{0}(s),\hat{n}_{0}(s)\}$ on $\mathbb{S}^{2}$ that follow from eq. $(\ref{5.38})$: unlike $\hat{n}^{\prime}(s)$ in eq. $(\ref{5.35})$, $\hat{n}^{\prime}_{0}(s)$ is not constant; and in detailed structure the present pair is different from that given by eq. $(\ref{5.30})$.

To sum up, in this Section we have analysed the structures of the BI $\Delta_{3}(\psi_{1},\psi_{2},\psi_{3})$ and of different kinds of NPC’s using the Schwinger–Majorana SU(2) framework. For any practical calculation of geometric phases along these lines, specific details will have to be worked out, but not involving any new points of principle.

In references 5, the reported values of geometric phases involve measuring directly phases of BI’s, not dealing with continuum Schrödinger evolution with some Hamiltonian along closed paths made up of geodesics in any state space. From our perspective in this and the previous Section, it would be interesting to design experiments involving continuous Schrödinger evolution along the sides of a ‘triangle’ in ray space, in which two sides (say) are geodesics while the third is a nontrivial (but as simple as possible) NPC.

V Concluding Remarks

The present work focusses on the various objects that appear in the equation

[TABLE]

relating the Pancharatnam phase and the geometric phase and invesigates their structure, properties, convenient parametrizations and useful decompositions.

We show that the third order BI $\Delta_{3}(\psi_{1},\psi_{2},\psi_{3})$ can be parametrised in terms of six unitary invariant angles. The algebraic independence or otherwise of these parameters depends on the dimension $n$ of the Hilbert space to which the vectors $\psi_{1},\psi_{2},\psi_{3}$ belong. With no specific assumptions regarding $\psi_{1},\psi_{2},\psi_{3}$ ( beyond those stipulated earlier) we show that for $n=2$ only five of these angle parameters are algebraically independent. On other hand for $n>2$ all six are algebraically independent. This difference in the two cases can be traced back to the fact that in a two dimensional Hilbert space at most two vectors can be linearly independent. As a curiosity we discuss in some detail the case $\psi_{1},\psi_{2},\psi_{3}$ are drawn from the set of coherent states of a one dimensional harmonic oscillator and find that, like the $n=2$ case, only five unitary invariant angles turn out to be algebraically independent although the underlying Hilbert space is infinite dimensional. This unexpected reduction may perhaps be traced back to interrelations implied by the overcompleteness of the set of coherent states. For $n=3$ we derive an explicit formula for the third order Bargmann invariant in terms of the intrinsic unitary invariant parameters. This, in turn, extends Pancharatnam’s result for the geometric phase $\varphi_{geom}$ pertaining to a spherical triangle on $\mathbb{S}^{2}$ to the corresponding case for $n=3$.

We give an explicit construction for the lifts of the null phase curves $N$ appearing in the BI-geometric phase connection above. Combining the ideas presented in ref. [5] on the use of Majorana representation 6 for symmetric quantum states with Schwinger’s work 7 on quantum theory of angular momentum, we develop what we call the Schwinger-Majorana framework for describing states of an $n$ level system in which the SU(2) group plays a key role. We use this framework to develop elegant and convenient descriptions for the third order BI and the lifts of the null phase curves and recover the results in ref. [5] expressing the general $n$ level system geometric phase as a sum of $(n-1)$ geometric phases of Pancharatnam type computed using $(n-1)$ geodesic triangles on the Poincaré sphere.

The three vertex Bargmann invariant plays an important role in quantum information theory in the context of distinguishability of three quantum states 12 ,13 ,14 and we expect that their description in the Schwinger-Majorana framework as developed here will find useful applications in this area. We further hope that our work will stimulate experimental activity in designing new experiments on geometric phases in higher dimensional systems based on direct measurement of the phase of the three vertex Bargmann invariant 5 ,15 as well those involving evolution along selected NPC’s generated by suitable Hamiltonians.

Acknowledgements.

KSA thanks the University Grant Commission for providing BSR-RFSMS fellowship. NM thanks the Indian National Science Academy for enabling this work through the INSA Distinguished Professorship.

Appendix A The Schwinger–Majorana framework for SU(2)

The three-parameter group SU(2) is the only compact simple Lie group which has one unitary irreducible representation (UIR), upto unitary equivalence, in every finite dimension $n=1,2,3,\ldots$. These UIR’s are labelled by the spin or angular momentum quantum number $J=0,1/2,1,\ldots$, with $n=2J+1$. The regular representations of SU(2) contain each UIR $J$ as often as its dimension $(2J+1)$. A much ‘leaner’ and very useful unitary representation of SU(2),the Schwinger representation arising from the Schwinger oscillator operator construction of the SU(2) Lie algebra 7 , has the attractive feature that it is the direct sum of all the UIR’s of SU(2), each occurring exactly once. This leads to a specific ‘model’ of finite dimensional Hilbert spaces $\mathcal{H}_{n}$ of all dimensions $n=1,2,3,\ldots$, once each, with certain common operator and vector features arising from SU(2) representation theory.

It has been recently pointed out that, thanks to the Majorana theorem for symmetric multispinor UIR’s of SU(2), there is a very natural framework to discuss geometric phases for quantum systems in any finite dimension 5 . We combine the Schwinger and Majorana ideas in this Appendix and recall the main features which are used in Section IV of the main text.

The Schwinger construction

This is based on two independent quantum mechanical oscillators with operators obeying the canonical commutation relations

[TABLE]

The hermitian SU(2) generators $\hat{J}_{j}$ and their commutation relations are

[TABLE]

The infinite dimensional Hilbert space $\mathcal{H}^{(Sch)}$ carrying an irreducible representation of eqs $(\ref{A.1})$ is the direct sum of finite dimensional subspaces $\mathcal{H}^{(Sch)}_{n}$, one for each $n=1,2,3,\ldots$ and mutually orthogonal:

[TABLE]

The subspace $\mathcal{H}^{(Sch)}_{n}$ carries the spin $J=\frac{1}{2}(n-1)$ UIR of SU(2), and can be used as a ‘model’ for an $n$-level quantum system. An ONB for it is given by

[TABLE]

When convenient, $|J,M\rangle$ will be written as $|n_{1},n_{2}\rangle$ with $n_{1}=J+M,n_{2}=J-M$, both integral:

[TABLE]

A general element $u\in SU(2)$ is unitarily represented on $\mathcal{H}^{(Sch)}$ as the exponential of $i$ times a real linear combination of $J_{j}$:

[TABLE]

Here $\boldsymbol{\alpha}$ are axis angle parameters; alternatively, using Euler angles, $D(u)$ is a product of three exponentials. On $\mathcal{H}^{(Sch)}_{n},D(u)$ reduces to the spin $J$ UIR $D^{(J)}(u)$ of SU(2). Of course, on a given $\mathcal{H}^{(Sch)}_{n}$ there is also action by $U(n)$ in its defining representation, containing $D^{(J)}(u)$.

The Majorana theorem, general vectors in $\mathcal{H}^{(Sch)}_{n}$

A general vector $|\psi\rangle$ in $\mathcal{H}^{(Sch)}_{n}$ is expressible in the ONB $(\ref{A.4})$ as a sum:

[TABLE]

Majorana’s theorem 6 is essentially the statement that $|\psi\rangle$ can also be expressed as the product of $(n-1)$ factors each linear in $\hat{a}_{\alpha}^{\dagger}$ acting on $|0,0\rangle$, apart from a constant factor:

[TABLE]

If some $\alpha_{k}$ vanish, the maximum of $M$ in $(\ref{A.7})$ is less than $J$; while if some $\beta_{k}$ vanish, the minimum of $M$ exceeds $-J$. For each $k$ we combine $\alpha_{k}$ and $\beta_{k}$ into a two-component complex column vector $\xi_{k}$ in a two-dimensional Hilbert space $\mathcal{H}_{2}$:

[TABLE]

Denote the collection $\xi_{1},\xi_{2},\ldots,\xi_{n-1}$ by $\{\xi\}$. Then define the (unnormalised) vectors

[TABLE]

with inner products

[TABLE]

From the use of the Poincaré–Bloch sphere $\mathbb{S}^{2}$ in polarization-spin problems, we know that each $\xi$ determines a point $\hat{n}\in\mathbb{S}^{2}$, while $\hat{n}$ determines $\xi$ upto a phase:

[TABLE]

If we define $z=\beta/\alpha$, we have the expressions

[TABLE]

As $|z|\rightarrow 0$ or $\infty,\hat{n}\rightarrow(0,0,1)$ or $(0,0,-1)$.

Returning to vectors $|\psi\rangle\in\mathcal{H}^{(Sch)}_{n}$, each factor in eq. $(\ref{A.8})$ leads to one point $\hat{n}_{k}\in\mathbb{S}^{2}$. Since the operator factors commute, these points are unordered. Thus each $|\psi\rangle$ leads to an unordered set $\{\hat{n}_{1},\hat{n}_{2},\ldots,\hat{n}_{2J}\}$ of $(n-1)$ points on $\mathbb{S}^{2}$; conversely the latter determines $|\psi\rangle$ upto a complex factor since

[TABLE]

Pure product vectors in $\mathcal{H}^{(Sch)}_{n}$

These are a subset of vectors in $\mathcal{H}^{(Sch)}_{n}$ which have a special property with respect to SU(2). They arise when in the unordered set $\{\xi\}=\{\xi_{1},\xi_{2},\ldots,\xi_{n-1}\}$, all the entries are the same. We introduce a simple notation for these vectors and can easily normalise them:

[TABLE]

Another easily obtained inner product is

[TABLE]

The special SU(2) related property of these vectors is that they are of highest weight, ‘$M=J$’:

[TABLE]

Thus they are SU(2) transforms of $|J,J\rangle$, hence an orbit under SU(2) action via the spin $J$ UIR in $\mathcal{H}^{(Sch)}_{n}$.

In summary we have three important results at the vector level in $\mathcal{H}^{(Sch)}_{n}$: the ONB $\{|J,M\rangle\}$; the representation of any $|\psi\rangle$ as a multiple of some $|\{\xi\};n\rangle$ for an unordered $\{\xi\}$; and the highest weight pure product states $|\xi;n\rangle$.