Self-Adjointness in Klein-Gordon Theory on Globally Hyperbolic Spacetimes

Abstract

We prove essential self-adjointness of the spatial part of the linear Klein-Gordon operator with external potential for a large class of globally hyperbolic manifolds. The proof is conducted by a fusion of new results concerning globally hyperbolic manifolds, the theory of weighted Hilbert spaces and related functional analytic advances.

Bosonic and fermionic Gaussian states from Kähler structures

“…which is proportional to the Killing form K(K + , K − ) on g, which we defined in (14). Therefore, the decomposition splits K over u(N ) and its orthogonal complement u ⊥ (N ), which corresponds exactly to this decomposition into complex linear algebra elements K − forming u(N ) and complex anti-linear algebra elements K + forming u ⊥ (N ).…”

“…Different Fock vacua are then related by Bogoliubov transformations [4][5][6]. We illustrate the relation between the formulation in terms of mode functions and the formulation in terms of Kähler structures discussed here and used in the context of quantum fields in curved spacetimes [10][11][12][13][14][15].…”

“…In the case of infinitely many degrees of freedom, two Gaussian states |J〉 and |J〉 give rise to unitarily equivalent Fock space representations if the Hilbert-Schmidt norm of J −J is finite. 14 Example 1 (Single mode pure Gaussian bosonic states). We consider a single bosonic mode withξ q,p ≡ (q,p) a,a † ≡ (â,â † ).…”

“…Consequently, there exist many reviews [1][2][3][7][8][9] focusing on specific applications of Gaussian states relevant for these research fields. Kähler structure were first used to describe Gaussian states in the context of quantum fields in curved spacetimes [10][11][12][13][14][15]. 1 Here, we assume 〈ψ|ξ a |ψ〉 = 0, but also treat the general case in section 3.…”

Observables in Quantum Mechanics and the Importance of Self-Adjointness

“…Self-adjointness and the theory of SA extensions are known to play important roles in a variety of physical contexts, including systems with a confined particle [21][22][23], Aharonov-Bohm effect [24][25][26][27], graphene [28], two and three dimensional delta function potentials [29], heavy atoms [30][31][32], singular potentials [33,34], Calogero models [35,36], anyons [37,38], anomalies [39][40][41], ζ-function renormalization [42], scattering theory [43], particle statistics [44], black holes [45][46][47][48][49], integrable system [50,51], Klein-Gordon equation [52], renormalons in QM [53], quasinormal modes [54], supersymmetric QM [55] and toy models for strings [56], spectral triple [57], noncommutative field theories [58][59][60], resolving the spacetime singularities [61][62][63][64][65] and even pl...…”