“Chiral Quantum Cloning” – Representation theory, spectral invariants and symmetries in algebraic conformal quantum field theory – Karl-Henning Rehren Institut f¨ ur Theoretische Physik, Universit¨ at G¨ ottingen “Mathematics and Quantum Physics”, Rome, July 8, 2013 KHR ”Mathematics and Quantum Physics” Chiral Quantum Cloning 1 / 34
What this talk is about PRELUDIO KHR ”Mathematics and Quantum Physics” Chiral Quantum Cloning 2 / 34
The world . . . . . . is two-dimensional. KHR ”Mathematics and Quantum Physics” Chiral Quantum Cloning 3 / 34
The chiral world . . . . . . is one-dimensional. “Time is Space”, 1 lightyear = 9.450.000.000.000.000 m KHR ”Mathematics and Quantum Physics” Chiral Quantum Cloning 4 / 34
The projective chiral world . . . . . . is a circle. KHR ”Mathematics and Quantum Physics” Chiral Quantum Cloning 5 / 34
The most important physical observables in Conformal QFT are CHIRAL QUANTUM FIELDS. Energy and momentum densities, charge densities, . . . KHR ”Mathematics and Quantum Physics” Chiral Quantum Cloning 6 / 34
Chiral quantum fields . . . . . . are assigned to points or intervals on the circle. KHR ”Mathematics and Quantum Physics” Chiral Quantum Cloning 7 / 34
“Chiral quantum cloning” Roberto’s canonical isomorphism Two copies of a quantum field in one interval = . . . . . . = one quantum field in two disjoint intervals. [Longo-Xu ’04, Kawahigashi-Longo ’05] KHR ”Mathematics and Quantum Physics” Chiral Quantum Cloning 8 / 34
What is such an isomorphism possibly good for? Applications: Study of representation theory NCG spectral invariants Multilocal symmetries Modular theory of two-intervals C 2 cofiniteness (?) KHR ”Mathematics and Quantum Physics” Chiral Quantum Cloning 9 / 34
TEMA CON VARIAZIONI KHR ”Mathematics and Quantum Physics” Chiral Quantum Cloning 10 / 34
The general setting AQFT on the circle Localized von Neumann algebras A ( I ) Local commutativity Diffeomorphism (= conformal) covariance Vacuum and positive-energy representations KHR ”Mathematics and Quantum Physics” Chiral Quantum Cloning 11 / 34
Recall: The split property . . . states that the map a ⊗ b �→ ab for a ∈ A ( I 1 ) and b ∈ A ( I 2 ) is an isomorphism of von Neumann algebras A ( I 1 ) ⊗ A ( I 2 ) and A ( I 1 ) ∨ A ( I 2 ) ≡ A ( I 1 ∪ I 2 ) (whenever I 1 and I 2 do not touch). . . . expresses the possibility of independent preparations of partial states on the subalgebras A ( I 1 ) and A ( I 2 ). . . . follows from a decent growth of the dimensions of the eigenspaces of L 0 (namely, e − β L 0 should be trace-class in the vacuum representation, Buchholz-Wichmann 1986). . . . is assumed throughout. KHR ”Mathematics and Quantum Physics” Chiral Quantum Cloning 12 / 34
Roberto’s canonical isomorphism . . . composes diffeomorphisms z → ±√ z with the split isomorphism: A ⊗ 2 ( I ) ≡ A ( I ) ⊗ A ( I ) Diff ⊗ Diff Split iso − → A ( I 1 ) ⊗ A ( I 2 ) − → A ( I 1 ) ∨ A ( I 2 ) [Longo-Xu ’04, Kawahigashi-Longo ’05] KHR ”Mathematics and Quantum Physics” Chiral Quantum Cloning 13 / 34
Chiral quantum cloning ι I = χ I 1 ∪ I 2 ◦ ( δ √· ⊗ δ −√· ) : A ⊗ 2 ( I ) → A ( I 1 ∪ I 2 ) , √ I and A ⊗ 2 ( I ) ≡ A ( I ) ⊗ A ( I ). where I 1 ∪ I 2 ≡ Some facts to memorize: These isomorphisms do not preserve the ground state (vacuum) because the tensor product suppresses all correlations, and because diffeomorphisms do not preserve the vacuum. They are defined for each interval I , but they are compatible as I increases. They extend to the entire circle minus a point. √· . Everything generalizes to n copies, n intervals, using n KHR ”Mathematics and Quantum Physics” Chiral Quantum Cloning 14 / 34
Mode-doubling For Virasoro and affine Kac-Moody algebras it is well-known that L n = 1 2 L 2 n + c n = 1 � � J a 2 J a and 16 δ n 0 2 n satisfy the same commutation relations as L n and J a n (with central charge 2 c resp. level 2 k ). This is “one half” of the canonical isomorphism, applying ι to T ( z 2 ) ⊗ 1 + 1 ⊗ T ( z 2 ) resp. J a ( z 2 ) ⊗ 1 + 1 ⊗ J a ( z 2 ). The “other half” requires half-integer modes, ie, is twisted. KHR ”Mathematics and Quantum Physics” Chiral Quantum Cloning 15 / 34
VAR 1: Representation theory Longo-Xu (2004) KHR ”Mathematics and Quantum Physics” Chiral Quantum Cloning 16 / 34
The canonical representation The canonical isomorphisms ι I = χ I 1 ∪ I 2 ◦ ( δ √· ⊗ δ −√· ) : A ⊗ 2 ( I ) → A ( I 1 ∪ I 2 ) extend to a representation π of A ⊗ 2 on S 1 \ {− 1 } = R . The extension to the entire circle S 1 is not possible. Instead π differs on both sides of z = − 1 by the flip a ⊗ b �→ b ⊗ a . ⇒ π is a soliton (or twisted) representation of A ⊗ 2 ( R ). π restricts to a true (DHR) representation π B of the flip-invariant subnet B on R . π B is reducible with exactly two irreducible components. KHR ”Mathematics and Quantum Physics” Chiral Quantum Cloning 17 / 34
Recall: Dimension and µ -index The dimension d π (possibly ∞ ) of a representation is the square root of the index of the subfactor π ( A ( I )) ⊂ π ( A ( I ′ )) ′ . [Longo 1989] The µ -index (possibly ∞ ) of a chiral CFT on S 1 is the index of the two-interval subfactor A ( I 1 ∪ I 2 ) ⊂ A (( I 1 ∪ I 2 ) ′ ) ′ in the vacuum representation. The µ -index equals the “global index” � d 2 µ A = π . π [Kawahigashi-Longo-M¨ uger 2001] KHR ”Mathematics and Quantum Physics” Chiral Quantum Cloning 18 / 34
The Longo-Xu dichotomy The CMS property: “ A has at most countably many sectors, all with finite dimension.” If A has the CMS property, then A ⊗ 2 and B ⊂ A ⊗ 2 have the CMS property. In this case, π B = π 1 ⊕ π 2 has finite dimension. Because the index of B ( I ) ⊂ A ⊗ 2 ( I ) is two, it follows that π has finite dimension. π = the index of π I ( A ⊗ 2 ( I )) ⊂ π I ′ ( A ⊗ 2 ( I ′ )) ′ equals the But d 2 two-interval µ -index of the net A . Thus, µ A is finite. ⇒ If A has at most countably many sectors, then either the number of sectors is actually finite, or some sector has infinite dimension. The first possibility also implies strong additivity, hence complete rationality. KHR ”Mathematics and Quantum Physics” Chiral Quantum Cloning 19 / 34
VAR 2: NCG spectral invariants Kawahigashi-Longo (2005) KHR ”Mathematics and Quantum Physics” Chiral Quantum Cloning 20 / 34
Recall: Modular theory Given a von Neumann algebra M and a faithful normal state ϕ = (Φ , · Φ), the polar decomposition of the antilinear operator m Φ �→ m ∗ Φ gives rise to an anti-involution J and a unitary 1-parameter group ∆ it such that JMJ = M ′ , σ t := Ad ∆ it | M ⊂ Aut ( M ) . The modular group σ t is an “intrinsic dynamics” of M , depending only on the state. + ) and Φ = Ω, one has ∆ it = e − 2 π ( L 1 − L − 1 ) t = For M = A ( S 1 scale transformation of R + (Bisognano-Wichmann property, geometric modular action), and J =CPT (Brunetti-Guido-Longo 1993). In particular, the M¨ obius group (in 4D: the Poincar´ e group) is “of modular origin”. KHR ”Mathematics and Quantum Physics” Chiral Quantum Cloning 21 / 34
The Kawahigashi-Longo state Non-commutative elliptic geometry interpretation of “modular” free energy and entropy. n intervals as n → ∞ unite discrete and conformal features of NCG. Consider the state ϕ n = ω ⊗ n ◦ ι − 1 n √ n on A ( I ), where ι n is the ( n -interval) canonical isomorphism, and ω ⊗ n = ω ⊗ · · · ⊗ ω the vacuum state of A ⊗ n ( I ). Its modular group is generated by − 2 π n ( L n − L − n ). Hence the entire diffeomorphism symmetry is of modular origin. One may extract spectral invariants (“entropies”). In particular, the µ -index arises as a spectral invariant. A chiral CFT is expected to live on the horizon of a Black Hole: Relation to Bekenstein entropy. KHR ”Mathematics and Quantum Physics” Chiral Quantum Cloning 22 / 34
VAR 3: Multilocal symmetries KHR-Tedesco (2013) KHR ”Mathematics and Quantum Physics” Chiral Quantum Cloning 23 / 34
Multilocal currents If A is the real free Fermi CFT, then A ⊗ 2 is the complex free Fermi CFT which has an SO (2) = U (1) local gauge symmetry γ : ( ψ 1 + i ψ 2 )( z ) �→ e i α ( z ) ( ψ 1 + i ψ 2 )( z ) . The generator of the gauge symmetry is a free Bose current affiliated with A ⊗ 2 ( I ). The canonical isomorphism maps the current J into the two-interval real Fermi algebra A ( I 1 ∪ I 2 ). The embedded current is “multi-local” : ι ( J ( z 2 )) = 1 2 z : ψ ( z ) ψ ( − z ): . KHR ”Mathematics and Quantum Physics” Chiral Quantum Cloning 24 / 34
Multilocal symmetries 1 The multilocal current 2 z : ψ ( z ) ψ ( − z ): generates transformations of the real Fermi field γ m = ι ◦ γ ◦ ι − 1 . γ m are multilocal symmetries = z -dependent SO (2) “mixings” between ψ ( z ) and ψ ( − z ). Similar with the stress-energy tensor of A ⊗ 2 (= generator of diffeomorphisms of A ⊗ 2 ). Embedded into A ( I 1 ∪ I 2 ), it generates diffeomorphisms plus mixing . KHR ”Mathematics and Quantum Physics” Chiral Quantum Cloning 25 / 34
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