A Variational Principle in Discrete Space-Time – Existence of Minimizers

We formulate a variational principle for a collection of projectors in an indefinite inner product space. The existence of minimizers is proved in various situations. In a recent book it was proposed to formulate physics with a new variational principle in space-time [2]. In the present paper we construct minimizers of this variational principle. In order to make the presentation self-contained and easily accessible, we introduce the mathematical framework from the basics (see Sections 1 and 2). Thus this paper can be used as an introduction to the mathematical setting of the principle of the fermionic projector. However, the reader who wants to get a physical understanding is referred to [2]. Our variational principle is set up in finite dimension, and thus the continuity of the action is not an issue. The difficulties are the lack of compactness and the fact that there is no notion of convexity. Therefore, we need to derive suitable estimates (Sections 4 and 5) before we can use the direct method of the calculus of variations (Sections 7 and 8). Our main results are stated in Section 2, whereas in Section 3 we explain our variational principle and illustrate it with a few simple examples. 1 Discrete Space-Time and the Fermionic Projector Let H be a finite-dimensional complex vector space, endowed with a sesquilinear form <.|.> : H ×H → C, i.e. for all u, v,w ∈ H and α, β ∈ C, = α + β <αu+ βv | w> = α + β . We assume that <.|.> is symmetric, = , and non-degenerate, = 0 ∀ v ∈ H =⇒ u = 0 . Note that <.|.> is in general not positive, and it is therefore not a scalar product. We also refer to (H,<.|.>) as an indefinite inner product space. To a non-degenerate subspace of H we can associate its signature (p, q), where p and q are the maximal dimensions of 1 positive and negative definite subspaces, respectively (for more details see [1, 3] and the examples in Section 3). Many constructions familiar from scalar product spaces can be carried over to indefinite inner product spaces. In particular, we define the adjoint of a linear operator A : H → H by the relation = ∀ u, v ∈ H . A linear operator A is said to be unitary if A∗ = A−1 and symmetric if A∗ = A. It is called a projector if it is symmetric and idempotent, A = A = A . Let M be a finite set. To every point x ∈ M we associate a projector Ex. We assume that these projectors are orthogonal and complete in the sense that Ex Ey = δxy Ex and ∑ x∈M Ex = 1 . (1) Equivalently, we can say that the images of the projectors Ex give a decomposition of H into orthogonal subspaces, H = ⊕ x∈M Ex(H) . (2) Furthermore, we assume that the images Ex(H) ⊂ H of these projectors are non-degenerate and all have the same signature (n, n). We refer to (n, n) as the spin dimension. Relation (2) shows that the dimension of H must be equal to m ·2n, where m = #M denotes the number of points of M . The points x ∈ M are called discrete space-time points, and the corresponding projectors Ex are the space-time projectors. The structure (H,<.|.>, (Ex)x∈M ) is called discrete space-time. We now introduce one more projector P on H, the so-called fermionic projector, which has the additional property that its image P (H) is negative definite. In other words, P (H) has signature (0, f) with f ∈ N. The vectors in the image of P have the interpretation as the quantum mechanical states of the particles of our system, and we call f = dimP (H) the number of particles. We remark that in physical applications [2] these particles are Dirac particles, which are fermions, giving rise to the name “fermionic projector”. A space-time projector Ex can be used to restrict an operator to the subspace Ex(H) ⊂ H. Using a more graphic notion, we also refer to this restriction as the localization at the space-time point x. For example, using the completeness of the space-time projectors (1), we readily see that f = TrP = ∑ x∈M Tr(ExP ) . (3) The expression Tr(ExP ) can be understood as the localization of the trace at the spacetime point x, and summing over all space-time points gives the total trace. We call Tr(ExP ) the local trace of P . When forming more complicated composite expressions in the projectors P and (Ex)x∈M , it is convenient to use the short notations P (x, y) = Ex P Ey and u(x) = Ex u . Referring to the orthogonal decomposition (2), P (x, y) maps Ey(H) to Ex(H) and vanishes otherwise. It is often useful to regard P (x, y) as a mapping only between these subspaces, P (x, y) : Ey(H) → Ex(H) . 2 Using (1), we can write the product Pu as follows, (Pu)(x) = Ex Pu = ∑