The number of orthogonal conjugations

After a short introduction to anti-linearity, bounds for the number of orthogonal (skew) conjugations are proved. They are saturated if the dimension of the Hilbert space is a power of two. For the other dimensions this is an open problem.


Introduction
The use of anti-linear or, as mathematicians call it, of conjugate linear operators in Physics goes back to E. P. Wigner, [12]. Wigner also discovered the structure of antiunitary operators, [13], in finite dimensional Hilbert spaces. The essential differences to the linear case is the existence of 2-dimensional irreducible subspaces so that the Hilbert space decomposes into a direct sum of 1-and 2-dimensional invariant spaces in general. Later on, F. Herbut and M. Vujičić could clarify the structure of anti-linear normal operators, [4], by proving that such a decomposition also exists for anti-linear normal operators. While any linear operator allows for a Jordan decomposition, I do not know a similar decomposition of an arbitrary anti-linear operator.
In the main part of the paper there is no discussion of what is happening in case of an infinite dimensional Hilbert space. There are, however, several important issues both in Physics and in Mathematics: A motivation of Wigner was in the prominent application of (skew) conjugations, (see the next section for definitions), to time reversal symmetry and related inexecutable symmetries. It is impossible to give credit to the many beautiful results in Elementary Particle Physics and in Minkowski Quantum Field Theory in this domain. But it is perhaps worthwhile to note the following: The CPToperator, the combination of particle conjugation C, parity operator P, and time-reversal T, is an anti-unitary operator acting on bosons as a conjugation and on fermions as a skew conjugation. It is a genuine symmetry of any relativistic quantum field theory in Minkowski space. The proof is a masterpiece of R. Jost, [7]. There is a further remarkable feature of anti-linearity shown by CPT. This operator is defined up to the choice of the point x in Minkowski space on which PT acts as x → −x. Calling this specific form CPT x , one quite forwardly shows that the linear operator CPT x CPT y is representing the translation by the vector 2(x − y).
The particular feature in the example at hand is the splitting of an executable symmetry operation into the product of two anti-linear ones. This feature can be observed also in some completely different situations. An example is the possibility to write the output of quantum teleportation, as introduced by Bennett et al [1], [8,2], as the action of the product of two anti-linear ones on the input state vector, see [9,10,3].
These few sketched examples may hopefully convince the reader that studying antilinearity is quite reasonable -though the topic of the present paper is by far not so spectacular.
The next two sections provide a mini-introduction to anti-linearity. In the last one it is proved that the number of mutually orthogonal (skew) conjugations is maximal if the dimension of the Hilbert space is a power of two. It is conjectured that there are no other dimensions for which this number reaches its natural upper bound.

Anti-(or conjugate) linearity
Let H be a complex Hilbert space of dimension d < ∞. Its scalar product is denoted by φ b , φ a for all φ a , φ b ∈ H. The scalar product is assumed linear in φ a . This is the "physical" convention going back to E. Schrödinger. 1 is the identity operator. Definition 1: An operator ϑ acting on a complex linear space is called anti-linear or, equivalently, conjugate linear if it obeys the relation As is common use, B(H) denotes the set (algebra) of all linear operators from H into itself. The set (linear space) of all anti-linear operators is called B(H) anti . Anti-linearity requires a special definition of the Hermitian adjoint. Definition 2 (Wigner): The Hermitian adjoint, ϑ † , of ϑ ∈ B(H) anti is defined by A simple but important fact is seen by commuting ϑ and A = c1. One obtains (cϑ) † = cϑ † , saying: ϑ → ϑ † is a complex linear operation, This is an essential difference to the linear case: Taking the Hermitian adjoint is a linear operation.
A similar argument shows, that the eigenvalues of an anti-linear ϑ form circles around zero. If there is at least one eigenvalue and d > 1, let r be the radius of the largest such circle. The set of all values φ, ϑφ , φ running through all unit vectors, is the disk with radius r. • See [6] for the more sophisticated real case.
We need some further definitions. Definition 3: An anti-linear operator ϑ is said to be Hermitian or self-adjoint if ϑ † = ϑ. ϑ is said to be skew Hermitian or skew self-adjoint if ϑ † = −ϑ. The linear space of all Hermitian (skew Hermitian) anti-linear operators are denoted by Rank-one linear operators are as usually written and we define similarly projecting any vector φ onto a multiple of φ . Remark that we do not use φ | decoupled from its other part. We do not attach any meaning to φ | anti as a standing alone expression 1 ! An anti-linear operator θ is called a unitary one or, as Wigner used to say, an antiunitary, if θ † = θ −1 . A conjugation is an anti-unitary operator which is Hermitian, hence fulfilling θ 2 = 1. The anti-unitary θ will be called a skew conjugation if it is skew Hermitian, hence satisfying θ 2 = −1.

The invariant Hermitian form
While the trace of an anti-linear operator is undefined, the product of two anti-linear operators is linear. The trace (ϑ 1 , ϑ 2 ) := Tr ϑ 2 ϑ 1 will be called the canonical Hermitian form, or just the canonical form on the the space of anti-linear operators. An anti-linear ϑ can be written uniquely as a sum ϑ = ϑ + + ϑ − of an Hermitian and a skew Hermitian operator with Relying on (5) and (6) one concludes In particular, equipped with the canonical form, B(H) + anti becomes an Hilbert space. Completely analogue, −(., .) is a positive definite scalar product on B(H) − anti . Bases of these two Hilbert spaces can be obtained as follows: Let φ 1 , φ 2 , . . . be a basis of H. Then where j.k = 1, . . . , d and k < j, is a basis of B(H) + anti with respect to the canonical form. As a basis of B(H) − anti one can use the anti-linear operators By counting basis lengths one gets It follows: The signature of the canonical Hermitian form is equal to d = dim H. Indeed, 4 Orthogonal (skew) conjugations Their scalar products are restrictions of the canonical form (up to a sign in the skew case). Therefore to ask for the maximal number of mutually orthogonal conjugation or skew conjugations, is a legitim question.
These two numbers depend on the dimension d = dim H of the Hilbert space only. Let us denote by N + (d) the maximal number of orthogonal conjugations and by N − (d) the maximal number of skew conjugations. By (10) it is To get an estimation from below, one observes that the tensor products of two conjugation and that of two skew conjugations are conjugations. Therefore and, similarly, because the direct product of two orthogonal (skew) conjugations is orthogonal. Now consider the case that equality holds in (12) for d 1 and d 2 . Then one gets the inequality and its right hand side yields d(d + 1)/2 with d = d 1 d 2 . Hence there holds equality in (13). A similar reasoning shows equality in (14) if equality holds in (12). Hence: The set of dimensions for which equality takes place in (12) is closed under multiplication.
To rephrase this result we call N anti the set of dimensions for which equality holds in (12): If d 1 ∈ N anti and d 2 ∈ N anti then d 1 d 2 ∈ N anti . 2 ∈ N anti will be shown by explicit calculations below. Hence every power of two is contained in N anti .
Let us shortly look at dim H = 1. It is N + (1) = 1 and N − (1) = 0. Indeed, any anti-linear operator in C is of the form ϑ a z = az * . This is a conjugation if |a| = 1. There are no skew conjugations. The canonical form reads (ϑ a , ϑ b ) = a * b. Conjecture: N anti consists of the numbers 2 n , n = 0, 1, 2, . . ..
Skew Hermitian invertible operators exist in even dimensional Hilbert spaces only. Therefore, no odd number except 1 is contained in N anti . This, however, is a rather trivial case. Already for dim H = 3 the maximal number N + (3) of orthogonal conjugations seems not to be known.
acknowledgements I like to thank B. Crell for helpful support.