A Note on Causality in the Lorentzian Moyal Plane Luca Tomassini Dipartimento di Economia, Universit`a di Chieti-Pescara “G. d’Annunzio” Viale Pindaro, 42, I-65127 Pescara, Italy E-mail:
[email protected]
Abstract In this note we show that by using the techniques of [11] the proofs and results in [10] concerning the casual relation between coherent states in the Lorentzian Moyal Triple can be extended to (translated of) arbitrary smooth states.
Work supported by the ERC advanced grant 669240 QUEST “Quantum Algebraic Structures and Models” and by GNAMPA-INDAM.
1
Contents 1 Introduction
3
2 Notations and Conventions
3
3 The Moyal Lorentzian Spectral Triple
4
4 Casual relations between translated states
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2
1
Introduction
The generalisation of noncommutative geometry `a la Connes [2] from riemannian to lorentzian manifolds has been pursued by several authors [3, 4, 5, 6, 12, 15, 16]. In particular, aside from the metric properties, the casual structure of lorentzian noncommutative spacetimes has deserved a strong attention (see for example [7, 8, 9, 10]). Not only it was shown that the algebraic definitions provide the usual metric and casual structures in the case of commutative globally hyperbolic spacetimes, but also for several noncommutative models the casual structure was extensively investigated. In particular, the Lorentzian Moyal Plane was considered in [10] were it was proved that coherent states (viewed as translates of the ground state) have the same casual relations between them as “commutative points”. In Section 2 we start by providing the basic definitions and then in Section 3 we review some well known facts about the Moyal Plane. In particular, we introduce light-cone coordinates, which appear particularly suitable in this context, and present some basic facts about translations. Finally, in Section 4 we present our main result. Essentially, it is a simple adaptation of the arguments in [10].
2
Notations and Conventions
We start by barely recalling the definition of Lorentzian Spectral Triple and of the corresponding casual relation between states as presented in [10] (see also [6]). e π, H, D, J) with: Definition 2.1. A Lorentzian spectral triple is given by (A, A, • A Hilbert space H. • A non unital pre-C*-algebra A with a faithful *-representation π on B(H). e of A, which is also a pre-C*-algebra, with a compatible • A preferred unitization A e faithful *-representation π on B(H) and such that A is an ideal of A. • An unbounded operator D, densely defined on H, such that: e [D, π(a)] extends to a bounded operator on H, – ∀a ∈ A, 1
– ∀a ∈ A, π(a)(1 + hDi2 )− 2 is compact, with hDi2 := 21 (DD∗ + D∗ D). e and • A bounded operator J on H with J 2 = 1, J ∗ = J, [J, π(a)] = 0, ∀a ∈ A such that: – D∗ = −JDJ on Dom(D) = Dom(D∗ ) ⊂ H; – there is a densely defined self-adjoint operator T with Dom(T) ∩ Dom(D) −1 e and a positive element N ∈ A e such dense in H and with (1 + T 2 ) 2 ∈ A, that J = −N [D, T]. 3
Definition 2.2. We say that a Lorentzian spectral triple is even if there exists a Z2 e γJ = −Jγ and grading γ of H such that γ ∗ = γ, γ 2 = 1, [γ, π(a)] = 0 ∀a ∈ A, γD = −Dγ. e Definition 2.3 ([7]). Let C be the convex cone of all Hermitian elements a ∈ A respecting ∀ φ ∈ H, hφ, J[D, π(a)]φi ≤ 0, (1) where h·, ·i is the inner product on H. If the following condition is fulfilled: e spanC (C) = A,
(2)
then C is called a causal cone. It induces a partial order relation on S(A), which we call causal relation, by: ∀ω, η ∈ S(A),
3
ωη
iff
∀a ∈ C,
ω(a) ≤ η(a).
(3)
The Moyal Lorentzian Spectral Triple
e π, H0 , D, J) on the Minkowski space R1,1 The Moyal Lorentzian spectral triple (A, A, is constructed in the following way [10]: • H0 := L2 (R1,1 ) ⊗ C2 is the Hilbert space of square integrable sections of the spinor bundle over the two-dimensional RMinkowski space-time with the usual positive definite inner product hψ, φi = d2 x (ψ1∗ φ1 + ψ2∗ φ2 ) ∀ ψ, φ ∈ H0 with ψ = (ψ1 , ψ2 ), φ = (φ1 , φ2 ). • A is the space of Schwartz functions S(R1,1 ) with the Moyal ? product Z 1 d2 s d2 t f (x + s) g(x + t) e−2iσ(s,t) , f, g ∈ S(R2 ). (4) (f ? g)(x) := 2 π R4 where σ(·, ·) denotes the standard symplectic form. The representation π : A → B(H0 ) is defined by the left multiplication: π(f ) = L(f ) ⊗ I2 ,
π(f )ψ = (L(f )ψ1 , L(f )ψ2 ) = (f ? ψ1 , f ? ψ2 ),
(5)
is faithful and its norm closure is isomorphic to the compact operators K(H0 ). Thus we will identify states on A and π(A). Moreover, any pure state ω ∈ S(A) is a vector state [14]. This means that there is a vector ψ ∈ H0 such that ω(f ) = hψ, π(f )ψi for all f ∈ A. • To simplify our discussion, we take the point of view [10] and will not choose as our preferred unitisation the unital Fr´echet pre-C*-algebra of smooth functions e⊂ which are bounded together with all derivatives (B, ?) but rather a larger A M(A), where M(A) = {T ∈ S 0 (R1,1 ) | T ?h ∈ S(R1,1 ) and h?T ∈ S(R1,1 ) for all h ∈ S(R1,1 )}, 4
in the multiplier algebra. In particular, it will include (linear combinations of powers of) unbounded “coordinate operators” (see below). However, we will e in their pay the price of being forced to restrict our attention to states having A domain. • D := −i∂µ ⊗ γ µ (with µ = 0, 1) is the flat Dirac operator on R1,1 where: 0 i 0 i 0 1 1 2 γ = iσ = , γ =σ = i 0 −i 0
(6)
are the flat Dirac matrices which verify γ µ γ ν + γ ν γ µ = 2η µν , ∀ µ, ν = 0, 1 (we comply with [10] and use (−, +) for the signature of the metric). • J := iγ 0 is the fundamental symmetry which turns the Hilbert space H0 into a Krein space. It is customary to consider1 “cartesian” generators (coordinates) x0 , x1 for the algebra A, with corresponding derivatives ∂1 , ∂2 . However, it is particularly useful to introduce light-cone coordinates and derivatives x0 − x1 x 0 + x1 x− := √ , (7) x+ := √ , 2 2 ∂0 + ∂1 ∂0 − ∂1 ∂+ := √ , ∂− := √ . (8) 2 2 Then, we can write the Dirac operator as √ 0 ∂+ D= 2 , (9) ∂− 0 The commutator of D with a Schwartz function f acts by √ √ 0 L(∂+ f ) ∂+ f ? ψ 2 . ψ= 2 [D, π(f )] ψ = 2 ∂− f ? ψ 1 L(∂− f ) 0
(10)
Moreover, the operator J[D, π(a)] of the causal constraint (1) is J[D, π(f )] = JDπ(f ) − Jπ(f )D √ −∂− L(f ) + L(f )∂− 0 = 2 0 −∂+ L(f ) + L(f )∂+ √ L(∂− f ) 0 = − 2 , 0 L(∂+ f ) and the condition is equivalent to Z Z 2 1,1 2 ∗ ∀ψ1 ∈ L (R ), d x ψ1 ((∂− f ) ? ψ1 ) = d2 x ψ1∗ ? (∂− f ) ? ψ1 ≥ 0, and 2
1,1
∀ψ2 ∈ L (R ), 1
Z
2
dx
ψ2∗ ((∂+ f )
Z ? ψ2 ) =
See below for some functional analytic details.
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d2 x ψ2∗ ? (∂+ f ) ? ψ2 ≥ 0.
(11)
(12)
(13)
3.1
Coordinates and unbounded operators
As already mentioned, we will especially need light-cone coordinate operators x+ , x− . Actually, they do not belong to (B, ?), so that we are led to consider unbounded operators. The essential fact here is that the Moyal product can be extended by continuity to the multiplier algebra M(A), which in turn contains the coordinate operators [1]. However, we need to be a bit more precise about domains. To this end, −ix1 +ix1 , z¯ = x0√ and consider Wigner’s transition eigenfunctions [1] set z = x0√ 2 2 r 2 −(x20 +x21 ) 1 hmn := √ z¯?m ? h00 ? z ?n , m, n ∈ N, h00 = e . (14) π m! n! They form an orthonormal basis of L2 (R1,1 ): δnp hmn ? hpq = √ hmq , 2π
h∗mn = hnm ,
(hmn , hkl ) = δmk δnl .
(15)
The point is that the linear span D of the hmn ’s for m, n ∈ N constitutes an invariant dense domain of analytic vectors for the symmetric operators L(x+ ), L(x− ) (or L(x0 ), L(x1 )) whose action writes √ √ ¯ m hm−1,n + λ m + 1 hm+1,n , L(x+ ) hmn = λ (16) √ √ ¯ L(x− ) hmn = λ m hm−1,n + λ m + 1hm+1,n , (17) ¯ = 1−i √ and λ √ . By virtue of a theorem of Nelson [13], all these operators with λ = 1+i 2 2 are essentially self-adjoint on D (i.e. D is a core for them all). Since D ⊂ S(R1,1 ) ⊂ L2 (R1,1 ), S(R1,1 ) is as well a core for all of them. From now on, all operators will be considered as defined on S(R1,1 ). On this domain, using i∂2 f , (18) x 0 ? f = x0 f + 2 and similar equations for x1 ? f and f ? xi (see [1]), one obtains a representation of the Heisenberg algebra: [L(x0 ), L(x1 )] = iI,
3.2
[L(x− ), L(x+ )] = iI.
(19)
Translations
Since we are modeling a noncommutative Minkowski space, it is natural to consider translations, that is the transformations (ακ f )(x) := f (x + κ)
(20)
with f ∈ S(R1,1 ) and κ, x ∈ R1,1 . Obviously fκ := ακ f is Schwartz and fκ ? gκ (x) = (f ? g)κ (x), so that ακ is a ∗-automorphism of the algebra A. In the left-regular representation, a translation by (κ0 , κ1 ) (respectively (κ+ , κ− ) in light-cone coordinates) 6
is implemented by the adjoint action of the plane wave with wave vector (−κ1 , κ0 ) (respectively (κ− , −κ+ )): for f ∈ S(R1,1 ), κ ∈ R1,1 , L(ακ f ) = Ad Uκ L(f ),
Uκ (x) := L(ei(−x0 κ1 +x1 κ0 ) ) = L(ei(x+ κ− −x− κ+ ) ),
(21)
where for simplicity we indicate by f (x) a function f ∈ S(R1,1 ) and not a function of the coordinate operators. One has, as operators on S(R1,1 ), L(κ · ∂f ) = i [L (−x0 κ1 + x1 κ0 ) , L(f )] = i [L (x+ κ− − x− κ+ ) , L(f )] ,
(22)
√ L(∂± f ) = ∓ 2i [L (x∓ ) , L(f )] ,
(23)
so that
The action Ad Uκ extends naturally to the multiplier algebra M(A) and we have, as operators on S(R1,1 ), the relations d d L (αtκ (f ))|t = L αtκ (f )|t = L (αtκ (κ+ ∂+ f + κ− ∂− f )) , (24) dt dt d d L (αtκ (x± ))|t = L αtκ (x± )|t = κ± I. (25) L (ακ (x± )) = L (x± + κ± ) , dt dt Definition 3.1.
• For κ ∈ R1,1 , the κ-translated of a state ω ∈ S(A) is ωκ := ω ◦ ακ
(26)
where ακ is defined in (20). m n n ) < +∞ for x ) , ω(x x • We say a state ω ∈ S(A) is smooth whenever ω(xm − + + − any m, n ∈ N. Notice that any smooth state can be decomposed into a convex combination of pure states which will again be smooth. Moreover, in our representation π of A pure smooth states are given by ψ = (ψ1 , ψ2 ) ∈ H0 such that ψ1 , ψ2 ∈ S(R1,1 ). This follows from the fact that ψ1 , ψ2 must be in the domain of any positive power of x1 , x2 , ∂1 , ∂2 . As a consequence, we can evaluate smooth states on the identities (24),(25) and they remain true.
4
Casual relations between translated states
We can now state and prove our result. Proposition 4.1. Suppose ω is a any smooth state and let ωκ be its translated by κ ∈ R1,1 . Then these states are casually related with ω ωk if and only if κ ∈ V+ = {κ+ , κ− ≥ 0}, the closed forward light-cone.
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Proof. We closely follow the arguments in [10] and start by showing that under the stated assumptions for each a ∈ C we have ωκ (a) − ω(a) ≥ 0. Suppose first that ω is pure. From the Fundamental Theorem of Calculus we get Z 1 dt (k+ ωtκ (∂+ a) + k− ωtκ (∂− a)) , (27) ωκ (a) − ω(a) = 0
and the result follows immediately by the remark above, the characterisation (12) of the convex cone C and the facts that all pure states are vector states. Finally, as remarked earlier general smooth states are convex combinations of pure smooth ones. Conversely, we need to show that for any κ ∈ / V+ we can find a ∈ A˜ such that ωκ (a) − ω(a) ≤ 0. This means that at least one of κ+ , κ− is stricly negative and with no loss of generality we will suppose it is κ+ . Consider2 the functions a+ = x0 + x1 , a− = x0 − x1 ∈ C and observe that γ+ a+ + γ− a− ∈ C as long as γ+ , γ− ≥ 0. But since Z 1 1 ωκ (a+ ) − ω(a+ ) = dt k+ ωtκ (∂+ a) = k+ , (28) 2 0 Z 1 1 dt k− ωtκ (∂− a) = k− , (29) ωκ (a− ) − ω(a− ) = 2 0 we see that it is enough to choose γ+ sufficiently big.
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¯ and a− = λz + λ¯ ¯ z (compare [10]). Note that a+ = λ¯ z + λz
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