ON THE STRUCTURE OF PROJECTIVE GROUP REPRESENTATIONS IN QUATERNIONIC HILBERT SPACE

TERRY TAO MATHEMATICS DEPARTMENT, FINE HALL, PRINCETON UNIVERSITY, PRINCETON, NJ 08544, USA ANDREW C. MILLARD PHYSICS DEPARTMENT, JADWIN HALL, PRINCETON UNIVERSITY, PRINCETON, NJ 08544, USA

Abstract. A structure theorem concerning projective quaternionic representations is stated and proved. A corollary to the structure theorem is then proved, answering a question recently posed by Adler.

PACS codes 02.10.Tq, 02.20.Qs, 02.40.-k

1. Introduction

In a recent paper[1], Adler builds on some of the material in his book[2], specifi-

cally that which concerns general, multi-central and central quaternionic projective

representations. He asks if a multi-central representation can be constructed that

is not central or if it can be proved that a multi-central representation must always

be central. For a given Lie group G, the answers to these questions depend on

the connectedness of the group. If G is not connected, it is possible to construct a

2 TERRY TAOMATHEMATICS DEPARTMENT, FINE HALL,PRINCETON UNIVERSITY, PRINCETON, NJ 08544, USAANDREW C. MILLARDPHYSICS DEPARTMENT, JADWIN HALL,PRINCETON UNIVERSITY, PRINCETON, NJ 08544, USA

multi-central representation that is not central and we provide an example of such in this paper. Our main result, however, is to prove that if G is connected, then a multi-central must also be central. This is actually a corollary to a structure theorem that divides irreducible representations into three types according to the reality and commutativity of the projective phases.

2. Useful Terms and Common Themes

For a general introduction to quaternions and their application to quantum mechanics, we refer the reader to Adler’s book[2]. The quaternions include both real and complex numbers, and by a suitable categorisation the commutativity properties of the different types of quaternion can be concisely summarised. In this paper, we use the following categories in the proof of the structure theorem and of its corollary.

Definition 1. A real number has vanishing i, j and k coefficients.

Definition 2. A complex number has vanishing j and k coefficients. It is the most general α-symplectic quaternion.

Definition 3. A strictly complex number has vanishing j and k coefficients but nonvanishing i coefficient.

Definition 4. A skew-complex number has vanishing 1 and i coefficients. It is a purely β-symplectic quaternion.

ON THE STRUCTURE OF PROJECTIVE GROUP REPRESENTATIONS IN QUATERNIONIC HILBERT SPACE 3

Definition 5. A strictly quaternionic number has non-vanishing β-symplectic part.

It should be noted that these definitions are dependent on the choice of i, j and k, so the quaternionic basis must be specified before the above terms can be used. Other useful quaternionic quantities are phases. A phase is a real, complex or quaternionic number that has unit norm, in other words it is an element of, respectively, U(1, R), U(1, C) or U(1, H). A real phase must then be ±1 while a complex phase can be parametrised as the familiar exp(iθ). The connectedness of G is crucial to the structure theorem and is used several times in the proof. Typically we will deduce that, for each a in G, a certain quantity dependent on a must have one of two mutually exclusive “disconnected” properties, P or Q. We use disconnected to mean that it is impossible to have a smooth transition from P to Q. Since G is connected, either property P holds for all a or property Q holds for all a; there cannot be one region of G where P is satisfied and one where Q is since the regions’ boundary would be a discontinuity. To determine which property holds for all a, it then suffices to test which property holds for some particular a, such as the multiplicative identity 1. As a tool for applying the consequences of connectedness, it is useful to be able to perturb group elements by a small amount. If we perturb a group element a by a small amount, we write the new group element as aδa where δa is a group element in some neighbourhood of the identity; letting δa tend to 1 then defines a limiting

4 TERRY TAOMATHEMATICS DEPARTMENT, FINE HALL,PRINCETON UNIVERSITY, PRINCETON, NJ 08544, USAANDREW C. MILLARDPHYSICS DEPARTMENT, JADWIN HALL,PRINCETON UNIVERSITY, PRINCETON, NJ 08544, USA

process that generates elements tending to a.

3. The Structure Theorem and its Corollary

Let G be a connected Lie group with identity element 1, and suppose that V is a quaternionic Hilbert space with a privileged set of (quaternionic) orthonormal basis elements {|f>}. Let

U : G → U(V, H)

be a smooth assignment from a group element a to a unitary right-linear operator U_{a} on V. We say that U is a projective quaternionic representation if U_{1} = 1 and

Ω(b, a) = U_{ba}^{−1}U_{b}U_{a}

is diagonal on the basis {|f>} for all group elements b and a, that is

Ω(b, a) = ∑_{f}|f>ω(f, b, a)<f|

for some quaternionic phase ω(f, b, a) called the projective phase. The above definition of a representation is partially dependent on the choice of basis; a representation which is projective with respect to one basis need not be projective with respect to another. However, one can safely reray a basis

|f> ↦ |f>ϕ_{f}
ω(f, b, a) ↦ \overline{ϕ_{f}}ω(f, b, a)ϕ_{f} ,

where the ϕ_{f} are quaternionic phases, without changing the projective nature of the map U — this reraying affects the ω but not the Ω. Let us also make the technical

ON THE STRUCTURE OF PROJECTIVE GROUP REPRESENTATIONS IN QUATERNIONIC HILBERT SPACE 5

assumption that U is irreducible, in the sense that there does not exist a partition of the basis elements into two sub-bases with respect to which the operators U_{a} are block-diagonal for all group elements a. A reducible representation can easily be expressed as the direct sum of irreducible ones, so this is not a severe restriction. There are three obvious ways to make such a representation. The first way is to start with a genuine representation (so that

U_{ba} = U_{b}U_{a}

for all b and a); this is, of course, automatically a quaternionic projective representation in any basis. We can modestly generalise this by requiring only that the representation be real-projective, that is, so that

U_{ba} = ±U_{b}U_{a} .

A second way is to start with a complex projective representation on a complex space V_{C}, so that one has complex operators U_{a}^{C} such that

U_{ba}^{C}ω(b, a) = U_{b}^{C}U_{a}^{C}

for some complex phase ω(b, a), and extend it to a projective quaternionic representation on V_{C} ⊕ V_{C}j in the canonical manner. The third way is to start with a real projective representation, U_{a}^{R} on a real space V_{R} and tensor product it with an otherwise arbitrary phase map

σ : G → U(1, H)

6 TERRY TAOMATHEMATICS DEPARTMENT, FINE HALL,PRINCETON UNIVERSITY, PRINCETON, NJ 08544, USAANDREW C. MILLARDPHYSICS DEPARTMENT, JADWIN HALL,PRINCETON UNIVERSITY, PRINCETON, NJ 08544, USA

to obtain a quaternionic projective representation on the quaternionic space V_{R} ⊗ H. The following theorem states that these three methods are, up to a reraying, the only ways to create irreducible projective quaternionic representations. Structure Theorem. Let U be an irreducible projective representation of a connected Lie group G. There then exists a reraying of the basis |f> under which one of the following three possibilities must hold.

(1) U is a real projective representation (Type I). That is, ω(f, b, a) = ω(b, a) is independent of |f> and is equal to ±1 for each b and a.

(2) U is the extension of a complex projective representation (Type II). That is, the matrix elements <f|U_{a}|f′> are complex and ω(f, b, a) = ω(b, a) is independent of |f> and is a complex phase.

(3) U is the tensor product of a real projective representation and a quaternionic

phase (Type III). That is, there exists a decomposition

U_{a} = U_{a}^{R} ∑_{f}|f>σ_{a}<f| ,

where the unitary operator U_{a}^{R} has real matrix elements, σ_{a} is a quaternionic phase and

U_{ba}^{R} = ±U_{b}^{R}U_{a}^{R}

for all b and a. It should be noted that the three cases are not mutually exclusive; for example, a

one-dimensional Type I representation is a Type III representation with real σ_{a} and

ON THE STRUCTURE OF PROJECTIVE GROUP REPRESENTATIONS IN QUATERNIONIC HILBERT SPACE 7

a one-dimensional Type II representation is a Type III representation with complex σ_{a}. We use this structure theorem to answer Adler’s question. Let a multi-central projective representation be one in which Ω(b, a) commutes with U_{b} and U_{a} for all b and a, and let a central projective representation be one in which Ω(b, a) commutes with U_{c} for all a, b and c. Our answer is then as follows.

Corollary 1. Any multi-central quaternionic projective representation of a connected Lie group is central.

The requirement that G is connected is not redundant, as shown in the following example. Set G to be the union of two copies of the quaternionic phase group U(1, H) labelled by + and − respectively, with these labels combining under Z_{2}. Let U be the one-dimensional representation defined by U_{a_{+}} = a and U_{a_{−}} = 1 for all a ∈ U(1, H). The whole system is now a quaternionic projective representation of Z_{2} × U(1, H), which is not connected, the operators multiplying according to the table. By inspection, for each row Ω(b, a) commutes with U_{a} and with U_{b}, so the representation is multi-central, but Ω(b, a) does not generally commute with all U_{c}, so the representation is not central.

4. Proof of the Structure Theorem

Following Adler’s analysis[1], we begin with the crucial observation that

8 TERRY TAOMATHEMATICS DEPARTMENT, FINE HALL,PRINCETON UNIVERSITY, PRINCETON, NJ 08544, USAANDREW C. MILLARDPHYSICS DEPARTMENT, JADWIN HALL,PRINCETON UNIVERSITY, PRINCETON, NJ 08544, USA

Statement 1.

U_{a}^{−1}Ω(b, c)U_{a} = Ω(bc, a)^{−1}Ω(b, ca)Ω(c, a)

is unitary and diagonal for all a, b and c.

We may therefore write this unitary diagonal operator as

U_{a}^{−1}Ω(b, c)U_{a} = ∑_{f}|f>ν(f, a, b, c)<f| (1)

where the ν are quaternionic phases that are formed from the ω,

ν(f, a, b, c) = ω(f, bc, a)^{−1}ω(f, b, ca)ω(f, c, a) .

We multiply equation 1 on the left by U_{a} and take matrix elements to give ω(f, b, c)<f|U_{a}|f′> = <f|U_{a}|f′>ν(f′, a, b, c) . (2) If the underlying algebra was commutative and the matrix elements of U_{a} were invertible, we could cancel the matrix elements and conclude that the ω were independent of f. Since the quaternions are not commutative, we can only expect to show that the conjugacy class of the ω is independent of f. We shall use equation 2 for various values of f and f′ to derive a relationship that involves the ω but not the ν. However, it is necessary to first ensure that the matrix elements of U_{a} are invertible for suitable a, |f> and |f′>. This motivates the following. Definition 6. We say that two basis states |f_{1}> and |f_{2}> are directly connected if there exists a basis state |f_{3}> (which may be equal to |f_{1}> or |f_{2}> or both) and a

ON THE STRUCTURE OF PROJECTIVE GROUP REPRESENTATIONS IN QUATERNIONIC HILBERT SPACE 9

group element a such that <f_{1}|U_{a}|f_{3}> and <f_{2}|U_{a}|f_{3}> are both non-zero (and hence invertible).

The relationship of direct connectivity is symmetric but is not necessarily an equivalence relation. In order to form equivalence classes, we shall also define a completion of direct connectivity. Definition 7. We say that two basis states |f> and |f′> are indirectly connected if there exists a non-negative integer n and a sequence of basis states |f_{0}>, …,|f_{n}> with |f_{0}> = |f> and |f_{n}> = |f′> such that |f_{i}> and |f_{i+1}> are directly connected for all 0 ≤ i < n.

It can easily be verified that indirect connectivity is an equivalence relation. Note that two states which are directly connected are automatically indirectly connected. This allows us to propose the following. Proposition 1. All basis states are indirectly connected to each other.

Suppose this were not true. Then one can partition the basis states into two nonempty classes A and A′ such that no state in A is directly connected to any state in A′. To do this, pick any basis state |f_{0}> and let A be the set of all states indirectly connected to |f_{0}>; by the hypothesis that not all states are indirectly connected, A and its complement A′ are non-empty. Let V_{A} and V_{A′} be the subspaces of V spanned by the states of A and those of A′ respectively, so that V = V_{A} ⊕ V_{A′}. Choose a basis state |f> from A and a group element a. Then the Hilbert space state U_{a}|f> can

10 TERRY TAOMATHEMATICS DEPARTMENT, FINE HALL,PRINCETON UNIVERSITY, PRINCETON, NJ 08544, USAANDREW C. MILLARDPHYSICS DEPARTMENT, JADWIN HALL,PRINCETON UNIVERSITY, PRINCETON, NJ 08544, USA

have components from states in A or components from states in A′ but not both, since that would contradict the fact that A and A′ are not directly connected. Since |f> is normalised and since U_{a} is unitary, U_{a}|f> is also normalised and hence lies either in the unit sphere of V_{A} or in the unit sphere of V_{A′} for each a. When a is the multiplicative group identity 1, U_{a}|f> lies in (the unit sphere of) V_{A}; by the connectivity argument, U_{a}|f> lies in V_{A} for all a, and since |f> was an arbitrary basis state of A, this means that U maps V_{A} to V_{A}. Similarly it must map V_{A′} to V_{A′}. Hence U is reducible, contradicting the assumptions of the theorem, and the proposition is proved.

Corollary 2. For any basis states |f_{1}> and |f_{2}> there exists a quaternionic phase ϕ(f_{1}, f_{2}) such that

ω(f_{2}, b, c) = ϕ(f_{1}, f_{2})ω(f_{1}, b, c)\overline{ϕ(f_{1}, f_{2})} (3)

for all b and c.

Since |f_{1}> and |f_{2}> are indirectly connected by the previous proposition, it is sufficient to prove this corollary for the case when the states are directly connected and then to use induction. (The claim is trivial when the states are the same.) We may thus assume the existence of a group element a and a state |f_{3}> such that <f_{1}|U_{a}|f_{3}> and <f_{2}|U_{a}|f_{3}> are invertible. Substituting f_{1} for f and f_{3} for f′ and then f_{2} for f and f_{3} for f′ in equation 2 and eliminating the ν dependence yields

ON THE STRUCTURE OF PROJECTIVE GROUP REPRESENTATIONS IN QUATERNIONIC HILBERT SPACE 11

equation 3 with

ϕ(f_{1}, f_{2}) = \frac{<f_{2}|U_{a}|f_{3}>[<f_{1}|U_{a}|f_{3}>]^{−1}}{|<f_{2}|U_{a}|f_{3}>[<f_{1}|U_{a}|f_{3}>]^{−1}|} = \overline{ϕ(f_{2}, f_{1})} .

Corollary 2 can now be used to simplify the problem as follows. Choose a fixed basis state |f_{0}>; all other basis states may now be rerayed,

|f> ↦ |f>ϕ(f, f_{0}) ,

without changing the projective nature of the representation U. Under this reraying, equation 3 implies that

ω(f, b, c) ↦ ω(f_{0}, b, c) ;

in other words, ω(f, b, c) = ω(b, c) is independent of the basis state |f>. There are then three cases depending on the range of the ω. We list them here and then consider each in turn.

(1) The ω(b, c) are real for all b and c.

(2) The ω are not always real, but they commute with each other.

(3) There exist ω(b, c) and ω(b′, c′) which do not commute with each other. For the first case, since the ω must have unit magnitude, each is therefore ±1 — this is the first case of the structure theorem so the representation is of Type I. Another way to state the second case is to say that there exists a single reraying under which the ω are all complex. In this case, we must re-examine equation 2, which is an identity of the form pr = rq. Now the ν are each the product of three of the ω; in particular, the ν are also complex. We can then use the following fact.

12 TERRY TAOMATHEMATICS DEPARTMENT, FINE HALL,PRINCETON UNIVERSITY, PRINCETON, NJ 08544, USAANDREW C. MILLARDPHYSICS DEPARTMENT, JADWIN HALL,PRINCETON UNIVERSITY, PRINCETON, NJ 08544, USA

Quaternionic Identity 1. If p is strictly complex, q are complex and r is quaternionic such that

pr = rq ,

then either p = q, in which case r is also complex, or p = q^{∗}, in which case r is skew-complex.

(This is easily proved by decomposing all quantities into complex and skew-complex components.) Since it is assumed that ω(b, c) is not real for at least one pair b and c, this means that, for each a, U_{a} either has all complex matrix elements or all skew-complex matrix elements. However, since U_{a} never completely vanishes and is complex when a = 1, we can use the connectedness of G as before to show that U_{a} must be complex for all a — this is the second case of the structure theorem so the representation is of Type II. For the third case, we must again re-examine equation 2. Since the ω are independent of |f> and the ν are formed from the ω, the ν are also independent of |f>. Thus, whenever x is any matrix element of U_{a}, we have

ω(b, c)x = xν(a, b, c) . (4)

When x and y are two non-zero matrix elements of U_{a}, a double application of equation 4 gives, after some manipulation, that

ω(b, c)xy^{−1} = xν(a, b, c)y^{−1} = xy^{−1}ω(b, c) ;

ON THE STRUCTURE OF PROJECTIVE GROUP REPRESENTATIONS IN QUATERNIONIC HILBERT SPACE 13

in other words, xy^{−1} commutes with ω(b, c). Similarly, xy^{−1} commutes with ω(b′, c′). Now we use the following fact.

Quaternionic Identity 2. If p, q and r are quaternions such that r commutes with both p and q, but p and q do not themselves commute, then r must be real.

Hence, xy^{−1} is real for all pairs of non-zero matrix elements x and y of U_{a}. This is equivalent to saying that the non-zero matrix elements of U_{a} all have the same quaternionic phase, so we can separate U_{a} into the product of a real matrix U_{a}^{R} and a quaternionic phase matrix Σ_{a},

Σ_{a} = ∑_{f}|f>σ_{a}<f| ,

so that

U_{a} = U_{a}^{R}Σ_{a} = Σ_{a}U_{a}^{R} .

Since U_{a} and Σ_{a} are unitary, U_{a}^{R} must also be unitary. Repeatedly using the fact that real matrices commute with constant diagonal ones (we recall that the Ω are constant diagonal), we find that

(U_{ba}^{R})^{−1}U_{b}^{R}U_{a}^{R}

is simultaneously real unitary and constant diagonal and must therefore be plus or minus the identity. Thus we have achieved the third case of the structure theorem and the representation is of Type III. Since these are the only three possibilities for the ω, the structure theorem is proved.

14 TERRY TAOMATHEMATICS DEPARTMENT, FINE HALL,PRINCETON UNIVERSITY, PRINCETON, NJ 08544, USAANDREW C. MILLARDPHYSICS DEPARTMENT, JADWIN HALL,PRINCETON UNIVERSITY, PRINCETON, NJ 08544, USA

5. Proof of the Structure Theorem’s Corollary

We may again assume without loss of generality that the representation is irreducible, in which case the structure theorem applies. Representations of Type I and Type II are automatically central, so we immediately pass to consideration of Type III representations. Here we effectively have a one-dimensional projective quaternionic representation

a ↦ σ_{a} .

Since the real part U_{a}^{R} of the representation is real projective and commutes with constant diagonal operators, the multi-centrality condition reduces to the quaternionvalued contraint

[σ_{ba}^{−1}σ_{b}σ_{a}, σ_{a}] = [σ_{ba}^{−1}σ_{b}σ_{a}, σ_{b}] = 0 . (5) One immediate consequence of multi-centrality is

Statement 2. If any two of σ_{a}, σ_{b} and σ_{ba} commute, then all three commute.

Our task is to show that in fact σ_{ba}^{−1}σ_{b}σ_{a} commutes with all σ_{c}, meaning that the representation is also central. We may assume that the σ_{a} do not all commute with one another as if they did, the representation would be of Type II which is always central. This lack of total commutativity may be stated as follows.

Statement 3. For any choice of i, there is an a such that σ_{a} is strictly quaternionic and so does not commute with any strictly complex number.

ON THE STRUCTURE OF PROJECTIVE GROUP REPRESENTATIONS IN QUATERNIONIC HILBERT SPACE 15

In particular, this means that we can always find an a such that σ_{a} is not real. Given that there are some non-commuting phases, centrality is equivalent to the statement that

σ_{ba} = ±σ_{b}σ_{a} (6) for all a and b, in other words the representation must, after all, be of Type I. We begin by observing that from equation 5 and from quaternionic identity 2, we have that

Statement 4. σ_{ba} = ±σ_{b}σ_{a} is true whenever σ_{a} and σ_{b} do not commute.

This is actually a special case of a more general statement.

Proposition 2. Unless exactly two of σ_{a}, σ_{b} and σ_{ba} are real, it is true that σ_{ba} = ±σ_{b}σ_{a}.

In other words, if none of, one of, or all three of σ_{a}, σ_{b} and σ_{ba} are real, then equation 6 holds. By statement 4, we may begin by assuming that σ_{a} and σ_{b} commute and hence by statement 2, they also commute with σ_{ba}. We can then reray them to be complex, in which case, statement 3 allows us to find a c such that σ_{c} does not commute with any of σ_{a}, σ_{b} or σ_{ba}. If all three of σ_{a}, σ_{b} and σ_{ba} are real, then the proposition is trivially true, so we can assume that at most one of them is real. This gives four cases as follows.

(1) σ_{b}, σ_{a} and σ_{ba} are all strictly complex.

16 TERRY TAOMATHEMATICS DEPARTMENT, FINE HALL,PRINCETON UNIVERSITY, PRINCETON, NJ 08544, USAANDREW C. MILLARDPHYSICS DEPARTMENT, JADWIN HALL,PRINCETON UNIVERSITY, PRINCETON, NJ 08544, USA

(2) σ_{b} is real but σ_{a} and σ_{ba} are strictly complex.

(3) σ_{a} is real but σ_{b} and σ_{ba} are strictly complex.

(4) σ_{ba} is real but σ_{b} and σ_{a} are strictly complex. In the first case, we have from statement 4 that

σ_{cb} = ±σ_{c}σ_{b} ;

since this is the product of something strictly complex and something strictly quaternionic, it is also strictly quaternionic. Hence, by statement 4 again,

σ_{cba} = ±σ_{cb}σ_{a} = ±σ_{c}σ_{b}σ_{a} .

On the other hand, another application of statement 4 gives that

σ_{cba} = ±σ_{c}σ_{ba} ,

which on cancelling σ_{c} means that equation 6 holds and the proposition is true. In the second case, we again have from statement 4 that

σ_{cba} = ±σ_{c}σ_{ba} ,

so σ_{cba} is strictly quaternionic and does not commute with σ_{a}. Hence, by statement 2, none of σ_{cba}, σ_{cb} or σ_{a} can commute with each other and hence by statement 4,

σ_{cba} = ±σ_{cb}σ_{a} ;

therefore,

σ_{c}σ_{ba} = ±σ_{cb}σ_{a}

ON THE STRUCTURE OF PROJECTIVE GROUP REPRESENTATIONS IN QUATERNIONIC HILBERT SPACE 17

or, after rearrangement,

σ_{a}σ_{ba}^{−1} = ±σ_{cb}^{−1}σ_{c}.

However, the left hand side of this equation is the product of two strictly complex quantities and hence is complex while the right hand side commutes with σ_{c} by statement 2; thus, both sides must be equal to ±1 or, since σ_{b} is real, to ±σ_{b}. Hence equation 6 holds and the proposition is again true. The third and fourth cases are handled in a similar manner to the above treatment of the second case, although σ_{bac} is considered instead of σ_{cba} in the third case. We now develop an important corollary to this proposition by introducing the following definitions.

Definition 8. σ is locally constant at x if there is a neighbourhood V_{x} around x such that σ_{x} = σ_{y} for all y in V_{x}.

Definition 9. If σ_{x} is real and σ is locally constant at x, then σ_{x} is strongly real.

The corollary to proposition 2 is then as follows.

Corollary 3. Unless exactly two of σ_{a}, σ_{b} and σ_{ba} are strongly real, it is true that σ_{ba} = ±σ_{b}σ_{a}.

This is trivially true if σ_{a}, σ_{b} and σ_{ba} are all strongly real, since they are then real and equation 6 holds. Suppose, then, that neither σ_{a} nor σ_{b} are strongly real: then we can find (multiplicative) perturbations aδa and bδb of a and b for which σ_{aδa} and

18 TERRY TAOMATHEMATICS DEPARTMENT, FINE HALL,PRINCETON UNIVERSITY, PRINCETON, NJ 08544, USAANDREW C. MILLARDPHYSICS DEPARTMENT, JADWIN HALL,PRINCETON UNIVERSITY, PRINCETON, NJ 08544, USA

σ_{bδb} are not real so that proposition 2 applies. Letting δa and δb tend to 1 and using the continuity of σ, equation 6 continues to hold and hence the proposition is true. The other two cases are handled similarly. We wish to prove that equation 6 always holds, and we shall do this by showing that there is no a such that σ_{a} is strongly real, allowing us to apply corollary 3. A convenient origin for our argument is the multiplicative identity of G.

Proposition 3. σ_{1} is real but is not strongly real.

Since σ_{a} commutes with itself, statement 2 implies that σ_{1} commutes with all σ_{a}. Since not all σ commute, quaternionic identity 2 means that σ_{1} is real. We will now prove that σ_{1} is not strongly real. Suppose that σ_{1} is strongly real, so that

σ_{δa} = ±1 (7)

for all δa in the neighbourhood V_{1}. Let a be any group element such that σ_{a} is not real; by continuity, we can then find a neighbourhood V_{1}^{a} of 1 such that σ_{aδa} is also not real and is not equal to −σ_{a} for all δa in V_{1}^{a}. Then, by proposition 2, we may conclude that

σ_{aδa} = σ_{a}

for all δa in V_{1} ∩ V_{1}^{a} — in other words, σ is locally constant at a whenever σ_{a} is not real. We will now generate a contradiction to the supposition that σ_{1} is strongly real.

ON THE STRUCTURE OF PROJECTIVE GROUP REPRESENTATIONS IN QUATERNIONIC HILBERT SPACE 19

We use the connectedness of G to construct a trajectory from a, where σ_{a} is not real, to the identity,

{a(t) : t ∈ [0, 1]} ; a(0) = a , a(1) = 1 .

At t = 0, σ_{a(t)} = σ_{a} but at t = 1, σ_{a(t)} ≠ σ_{a}. Let t_{0} be the first time that σ_{a(t)} departs from σ_{a}; more precisely,

t_{0} = sup{s : σ_{a(t)} = σ_{a} ∀ 0 ≤ t < s} .

By continuity, we then have that σ_{a(t_{0})} = σ_{a} so that σ_{a(t_{0})} is not real. Now, by the argument in the previous paragraph, which we recall follows from the supposition that σ_{1} is strongly real, σ is locally constant at a. However, this means that

σ_{a(t_{0}+δt)} = σ_{a(t_{0})}

for all sufficiently small δt. However, this contradicts the definition of t_{0}, which means that σ_{1} cannot be strongly real, proving the proposition. Since σ_{1} is not strongly real, we ought to be able to prove the converse result of the conclusion of the second paragraph of the above proof, and this is in fact the case.

Corollary 4. If σ_{a} is not real, then σ is not locally constant at a.

If σ is locally constant at a, then for all aδa in the neighbourhood V_{a} of a,

σ_{aδa} = σ_{a} ;

20 TERRY TAOMATHEMATICS DEPARTMENT, FINE HALL,PRINCETON UNIVERSITY, PRINCETON, NJ 08544, USAANDREW C. MILLARDPHYSICS DEPARTMENT, JADWIN HALL,PRINCETON UNIVERSITY, PRINCETON, NJ 08544, USA

now, σ_{a} is not real and so by proposition 2,

σ_{aδa} = ±σ_{a}σ_{δa}

which, with the previous equation, means that equation 7 is again true. However, that implies that σ_{1} is strongly real, which contradicts the proposition.

Corollary 5. If at least two of σ_{a}, σ_{b} and σ_{ba} are not strongly real, then all three phases are not strongly real.

Suppose that σ_{a} and σ_{b} are not strongly real but that σ_{ba} is. Since σ_{a} is not strongly real, corollary 4 implies that σ is not locally constant at a. Thus we can find a δa close to 1 such that σ_{aδa} is not equal to ±σ_{a} or ±1; in particular σ_{aδa} is not strongly real, so by corollary 3,

σ_{b}σ_{aδa} = ±σ_{baδa} .

However, since σ_{ba} is strongly real,

σ_{baδa} = ±σ_{ba} = ±σ_{b}σ_{a} ,

contradicting the fact that σ_{aδa} ≠ ±σ_{a}. The other two cases are handled similarly.

Corollary 6. For δa sufficiently close to 1, σ_{δa} is not strongly real.

Choose a such that σ_{a} is not real. By continuity, σ_{aδa} is also not real for δa sufficiently close to 1. Since σ_{a} and σ_{aδa} are not real, they are not strongly real, so by corollary 5, σ_{δa} is also not strongly real.

ON THE STRUCTURE OF PROJECTIVE GROUP REPRESENTATIONS IN QUATERNIONIC HILBERT SPACE 21

With one main step remaining, we are now very close to our goal. Proposition 4. There is no a such that σ_{a} is strongly real.

Suppose there exists an a such that σ_{a} is strongly real. We use the connectedness of G to construct a trajectory from a to the identity,

{a(t) : t ∈ [0, 1]} ; a(0) = a , a(1) = 1 .

At t = 0, σ_{a(t)} = σ_{a} and hence is strongly real, but at t = 1, σ_{a(t)} = σ_{1} which is not strongly real. Let t_{0} be the first time that σ_{a(t)} is not strongly real; more precisely,

t_{0} = sup{s : σ_{a(t)} is strongly real ∀ 0 ≤ t < s} .

The definition of t_{0} means that σ_{a(t)} is strongly real for all t < t_{0} and that, for all ϵ > 0, we can find δt ∈ [0, ϵ] such that σ_{a(t_{0}+δt)} is not strongly real. Suppose σ_{a(t_{0})} was itself strongly real. There is then a neighbourhood V_{a(t_{0})} such that σ is real and constant in that neighbourhood. However, this lets us choose an ϵ > 0 such that a(t_{0} + δt) ∈ V_{a(t_{0})} for all δt ∈ [0, ϵ]; since σ is real and constant in this neighbourhood, this means that σ_{a(t_{0}+δt)} is also strongly real, contradicting the definition of t_{0}. Suppose, then, that σ_{a(t_{0})} is not strongly real. Choose δa near 1 such that

a(t_{0})δa = a(t_{0} − δt)

where δt is, as usual, small and positive. Since σ_{a(t_{0})} and, by corollary 6, σ_{δa} are not strongly real, by corollary 5, σ_{a(t_{0}−δt)} is not strongly real, again contradicting the

22 TERRY TAOMATHEMATICS DEPARTMENT, FINE HALL,PRINCETON UNIVERSITY, PRINCETON, NJ 08544, USAANDREW C. MILLARDPHYSICS DEPARTMENT, JADWIN HALL,PRINCETON UNIVERSITY, PRINCETON, NJ 08544, USA

definition of t_{0}.

Since σ_{a(t_{0})} must be either strongly real or not strongly real, the supposition that σ_{a} was strongly real is false and we have proved the proposition. By corollary 3, therefore, equation 6 is always true, so that the representation is central, in the case that the σ_{a} do not all commute. All cases now having been covered, we have also proved the main corollary.

An alternative to the last couple of steps is to form H, the set of all a ∈ G such that σ_{a} is not strongly real. H is then a subgroup of G by corollary 5. We then prove that H is an open subset of G, and that G\H, the set of all a ∈ G such that σ_{a} is strongly real, is also an open subset of G. However, by the definition of connectedness, there is no way to partition G as the union of two disjoint non-empty open sets, and since 1 is in H by proposition 3, it must be G\H that is empty, proving proposition 4.

The above proof of the corollary to the structure theorem shows that for a multicentral (and hence central) Type III representation, either all the σ_{a} commute and can thus be rerayed to be complex — so the representation is Type II — or the σ satisfy equation 6 — so the representation is Type I. The only irreducible multicentral projective quaternionic representations are, therefore, either real projective representations or extensions of complex projective representations.

Acknowledgements

The authors would like to thank S. L. Adler for suggesting we look into the question

of multi-centrality implying centrality and for his useful suggestions while reading

this paper as it developed.

References

1. S. L. Adler, “Projective Group Representations in Quaternionic Hilbert Space”, Journal of Mathematical Physics (in press)

2. S. L. Adler, “Quaternionic Quantum Mechanics and Quantum Fields” (Oxford University Press, New York, 1995)

a b ba U_{a} U_{b} U_{ba} Ω(b, a)
a_{+} b_{+} ba_{+} a b ba  1
a_{+} b_{−} ba_{−} a 1 1  a
a_{−} b_{+} ba_{−} 1 b 1  b
a_{−} b_{−} ba_{+} 1 1 ba  ba

Representation Table for Z_{2} × U(1, H)