Motivated by formal similarities between the continuum limit of the Ising model and the Unruh effect, this paper connects the notion of an Ishibashi state in boundary conformal field theory with the Tomita–Takesaki theory for operator algebras. A geometrical approach to the definition of Ishibashi states is presented, and it is shown that, when normalizable, the Ishibashi states are cyclic separating states, justifying the operator state corespondence. When the states are not normalizable Tomita–Takesaki theory offers an alternative approach based on left Hilbert algebras, making possible extensions of our construction and the state-operator correspondence.
REFERENCES
1.
Araki
, H.
, “On quasifree states of CAR and Bogoliubov automorphisms
,” Publ. Res. Inst. Math. Sci.
6
, 385
–442
(1970/71
).2.
Birke
, L.
, and Fröhlich
, J
., “KMS etc.
,” Rev. Math. Phys.
14
, 829
–871
(2002
).3.
Cardy
, J.L.
, “Effect of boundary conditions on the operator content of two-dimensional conformally invariant theories
,” Nucl. Phys. B
275
, 200
–218
(1986
).4.
Cardy, J.L., “Conformal invariance,” in Phase Transitions and Critical Phenomena, edited by C. Domb and J.L. Lebowitz (Academic, London, 1987), Vol. 11.
5.
Cardy
, J.L.
, “Boundary conditions, fusion rules and the Verlinde formula
,” Nucl. Phys. B
324
, 581
–596
(1989
).6.
Cardy
, J.L.
, and Lewellen
, D.C.
, “Bulk and boundary operators in conformal field theory
,” Phys. Lett. B
259
, 274
–278
(1991
).7.
Connes, A., Noncommutative Geometry (Academic, San Diego, 1994).
8.
van Daele
, A.
, “A new approach to the Tomita-Takesaki theory of generalised Hilbert algebras
,” J. Funct. Anal.
15
, 378
–393
(1974
).9.
Evans, D.E., and Kawahigashi, Y., Quantum Symmetries on Operator Algebras (Clarendon, Oxford, 1998).
10.
Ghoshal
, S.
, and Zamolodchikov
, S.
, “Boundary S matrix and boundary states in two-dimensional integrable quantum field theory
,” Int. J. Mod. Phys. A
9
, 3841
–3885
(1994
);11.
Glimm, J., and Jaffe, A., Quantum Physics (Springer, New York, 1981).
12.
Guillemin
, V.
, Quillen
, D.
, and Sternberg
, S.
, “The classification of the irreducible complex algebras of infinite type
,” J. Anal. Math.
18
, 107
–112
(1967
).13.
Isham, C. (private communication).
14.
Ishibashi
, N.
, “The boundary and crosscap states in conformal field theories
,” Mod. Phys. Lett. A
4
, 251
–264
(1989
).15.
Kadison, R.V., and Ringrose, J.R., Fundamentals of the Theory of Operator Algebras (Academic, London, 1986), Vol. II.
16.
Leclair
, A.
, Mussardo
, G.
, Saleur
, H.
, and Skorik
, S.
, “Boundary energy and boundary states in integrable quantum field theories
,” Nucl. Phys. B
453
, 581
–618
(1995
);17.
Longo
, R.
, “Simple injective subfactors
,” Adv. Math.
63
, 152
–171
(1987
).18.
Osterwalder
, K.
, and Schrader
, R.
, “Axioms for Euclidean Green’s functions. I, II
,” Commun. Math. Phys.
31
, 83
–112
(1973
);Osterwalder
, K.
, and Schrader
, R.
, Commun. Math. Phys.
42
, 281
–305
(1975
).19.
Powers
, R.
, and Størmer
, E.
, “Free states of the canonical anticommutation relations
,” Commun. Math. Phys.
16
, 1
–33
(1970
).20.
Plymen, R.J., and Robinson, P.L., Spinors in Hilbert Space, Cambridge Tracts in Mathematics Vol. 114 (Cambridge University Press, Cambridge, 1994).
21.
Semplice, M., “Boundary conformal fields and Tomita-Takesaki theory,” Ph.D. thesis, Oxford, 2002.
22.
Sewell
, G.
, “Quantum fields on manifolds: PCT and gravitationally induced thermal states
,” Ann. Phys. (N.Y.)
141
, 201
–224
(1982
).23.
Stratila, S., and Zsido, L., Lectures on von Neumann Algebras, English edition (Abacus, Tunbridge Wells, 1979).
24.
Takesaki, M., Tomita’s Theory of Madular Hilbert Algebras and its Applications, Lecture Notes in Mathematics Vol. 128 (Springer, Heidelberg, 1970).
25.
Unruh
, W.G.
, “Notes on black hole evaporation
,” Phys. Rev. D
14
, 870
–892
(1976
).26.
Wald, R.M., Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics (University of Chicago Press, Chicago, 1994).
27.
Wassermann, A.S., Seminar, Oxford, 2002.
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