The many classical faces of quantum structures Chris Heunen University of Oxford November 30, 2013 1 / 31
Relationship between classical and quantum 2 / 31
Relationship between classical and quantum 2 / 31
Relationship between classical and quantum 2 / 31
Quantum logic ❤❤❤❤❤❤❤ ✭ ✭✭✭✭✭✭✭ Subsets of a set ❤ Subspaces of a Hilbert space 3 / 31
Quantum logic ❤❤❤❤❤❤❤ ✭ ✭✭✭✭✭✭✭ Subsets of a set ❤ Subspaces of a Hilbert space orthomodular lattice 1 a ⊥ b ⊥ a b 0 3 / 31
Quantum logic ❤❤❤❤❤❤❤ ✭ ✭✭✭✭✭✭✭ Subsets of a set ❤ Subspaces of a Hilbert space orthomodular lattice not distributive � � or and � = � � � � and or and 4 / 31
Quantum logic ❤❤❤❤❤❤❤ ✭ ✭✭✭✭✭✭✭ Subsets of a set ❤ Subspaces of a Hilbert space orthomodular lattice not distributive biscuit coffee tea nothing 5 / 31
Quantum logic ❤❤❤❤❤❤❤ ✭ ✭✭✭✭✭✭✭ Subsets of a set ❤ Subspaces of a Hilbert space orthomodular lattice not distributive biscuit coffee tea nothing More problems: no good → , ∀ , ∃ , ⊗ 5 / 31
Quantum logic ❤❤❤❤❤❤❤ ✭ ✭✭✭✭✭✭✭ Subsets of a set ❤ Subspaces of a Hilbert space orthomodular lattice not distributive biscuit coffee tea nothing More problems: no good → , ∀ , ∃ , ⊗ However: fine when within Boolean block / orthogonal basis! 5 / 31
Part I Algebras of observables 6 / 31
Algebras of observables Observables are primitive, states are derived 7 / 31
Algebras of observables Observables are primitive, states are derived C*-algebras ∗ -algebra of operators that is closed AW*-algebras abstract/algebraic version of W*-algebra von Neumann algebras / W*-algebras ∗ -algebra of operators that is weakly closed 7 / 31
Algebras of observables Observables are primitive, states are derived C*-algebras ∗ -algebra of operators that is closed AW*-algebras abstract/algebraic version of W*-algebra von Neumann algebras / W*-algebras ∗ -algebra of operators that is weakly closed Jordan algebras JC/JW-algebras: real version of above 7 / 31
Classical mechanics ◮ If X is a state space, then C ( X ) = { f : X → C } is an operator algebra. 8 / 31
Classical mechanics ◮ If X is a state space, then C ( X ) = { f : X → C } is an operator algebra. Theorem : Every commutative operator algebra ◮ is of this form. 8 / 31
Classical mechanics ◮ If X is a state space, then C ( X ) = { f : X → C } is an operator algebra. Theorem : Every commutative operator algebra ◮ is of this form. ◮ Can recover states (as maps C ( X ) → C ): “spectrum” Constructions on states transfer to observables: X + Y �→ C ( X ) ⊗ C ( Y ) X × Y �→ C ( X ) ⊕ C ( Y Equivalence of categories: states determine everything 8 / 31
Quantum mechanics ◮ If H is a Hilbert space, then B ( H ) = { f : H → H } is an operator algebra. 9 / 31
Quantum mechanics ◮ If H is a Hilbert space, then B ( H ) = { f : H → H } is an operator algebra. Theorem : Every operator algebra ◮ embeds into one of this form. 9 / 31
Quantum mechanics ◮ If H is a Hilbert space, then B ( H ) = { f : H → H } is an operator algebra. Theorem : Every operator algebra ◮ embeds into one of this form. ◮ Recover states? Do states determine everything? “Noncommutative spectrum”? 9 / 31
Quantum state spaces? certain convex sets (states) sheaves over locales (prime ideals) quantales (maximal ideals) orthomodular lattices (projections) q-spaces (projections of enveloping W*-algebra) 10 / 31
✤ � � ✤ � ✤ � ✤ � Quantum state spaces? spectrum commutative state spaces ≃ operator algebras � � quantum operator algebras ❴ ❴ ❴ ❴ ❴ state spaces 11 / 31
✤ � � ✤ � ✤ � ✤ � Quantum state spaces? No! spectrum commutative state spaces ≃ operator algebras � � G quantum operator algebras ❴ ❴ ❴ ❴ ❴ state spaces F Theorem : If G is continuous, ◮ then F degenerates. 11 / 31
� ✤ � ✤ ✤ � � � ✤ Quantum state spaces? No! spectrum commutative state spaces ≃ operator algebras � � G quantum operator algebras ❴ ❴ ❴ ❴ ❴ state spaces F Theorem : If G is continuous, ◮ then F degenerates. That’s right. ( F ( M n ) = ∅ for n ≥ 3.) ◮ 11 / 31
✤ � � ✤ � ✤ � ✤ � Quantum state spaces? No!? spectrum commutative state spaces ≃ operator algebras � � G quantum operator algebras ❴ ❴ ❴ ❴ ❴ state spaces F Theorem : If G is continuous, ◮ then F degenerates. That’s right. ( F ( M n ) = ∅ for n ≥ 3.) ◮ ◮ So G better not be continuous So quantum state spaces must be radically different ... 11 / 31
Part II Classical viewpoints 12 / 31
Doctrine of classical concepts “However far the phenomena transcend the scope of classical physical explanation, the ac- count of all evidence must be expressed in classi- cal terms.... The argument is simply that by the word experiment we refer to a situation where we can tell others what we have done and what we have learned and that, therefore, the account of the experimental arrangements and of the re- sults of the observations must be expressed in unambiguous language with suitable application of the terminology of classical physics.” 13 / 31
Classical viewpoints ◮ Invariant that circumvents the obstruction: Given an operator algebra A , consider C ( A ) = { C ⊆ A commutative subalgebra } , the collection of classical viewpoints. 14 / 31
Classical viewpoints ◮ Invariant that circumvents the obstruction: Given an operator algebra A , consider C ( A ) = { C ⊆ A commutative subalgebra } , the collection of classical viewpoints. Theorem : Can reconstruct A as a piecewise algebra. ◮ ( A ∼ = colim C ( A )) 14 / 31
Piecewise structures A piecewise widget is a widget that forgot ◮ operations between noncommuting elements. 15 / 31
Piecewise structures A piecewise widget is a widget that forgot ◮ operations between noncommuting elements. ◮ A piecewise complex *-algebra is a set A with: ◮ a reflexive binary relation ⊙ ⊆ A 2 ; ◮ (partial) binary operations + , · : ⊙ → A ; ◮ (total) functions ∗ : A → A and · : C × A → A ; such that every S ⊆ A with S 2 ⊆ ⊙ is contained in a T ⊆ A with T 2 ⊆ ⊙ where ( T , + , · , ∗ ) is a commutative ∗ -algebra. 15 / 31
Piecewise structures A piecewise widget is a widget that forgot ◮ operations between noncommuting elements. ◮ A piecewise Boolean algebra is a set B with: ◮ a reflexive binary relation ⊙ ⊆ B 2 ; ◮ (partial) binary operations ∨ , ∧ : ⊙ → B ; ◮ a (total) function ¬ : B → B ; such that every S ⊆ B with S 2 ⊆ ⊙ is contained in a T ⊆ B with T 2 ⊆ ⊙ where ( T , ∧ , ∨ , ¬ ) is a Boolean algebra. 15 / 31
Piecewise structures A piecewise widget is a widget that forgot ◮ operations between noncommuting elements. ◮ Every projection lattice gives a piecewise Boolean algebra: • ❲ ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣ ❚ ❲ ❥❥❥❥❥❥❥❥❥❥❥❥❥ ❚ ❲ ❍ ❲ ❚ ❲ ❍ ❚ ✈✈✈✈✈✈ ❲ ❚ ❲ ❍ ❲ ❚ ❲ ❚ ❲ ❍ ❲ ❚ ❲ ❲ ❍ ❚ ❲ ❚ ❲ ❍ ❚ ❲ ❚ ❲ ❲ ❚ ❲ • • • • • ❲ • ❍ ❍ ❍ ❍ ✈✈✈✈✈✈ ❍ ✈✈✈✈✈✈ ❍ ✈✈✈✈✈✈ ❍ ✈✈✈✈✈✈ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ • • • • • • ❲ ❲ ❚ ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣ ❲ ❚ ❥❥❥❥❥❥❥❥❥❥❥❥❥ ❲ ❍ ❚ ❲ ❲ ❚ ❍ ✈✈✈✈✈✈ ❲ ❚ ❲ ❍ ❲ ❚ ❲ ❚ ❲ ❍ ❲ ❚ ❲ ❚ ❍ ❲ ❚ ❲ ❲ ❚ ❍ ❲ ❚ ❲ ❲ ❚ ❲ • 15 / 31
Piecewise structures A piecewise widget is a widget that forgot ◮ operations between noncommuting elements. ◮ Every projection lattice gives a piecewise Boolean algebra: • ❲ ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣ ❚ ❲ ❥❥❥❥❥❥❥❥❥❥❥❥❥ ❚ ❲ ❍ ❲ ❚ ❲ ❍ ❚ ✈✈✈✈✈✈ ❲ ❚ ❲ ❍ ❲ ❚ ❲ ❚ ❲ ❍ ❲ ❚ ❲ ❲ ❍ ❚ ❲ ❚ ❲ ❍ ❚ ❲ ❚ ❲ ❲ ❚ ❲ • • • • • ❲ • ❍ ❍ ❍ ❍ ✈✈✈✈✈✈ ❍ ✈✈✈✈✈✈ ❍ ✈✈✈✈✈✈ ❍ ✈✈✈✈✈✈ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ • • • • • • ❲ ❲ ❚ ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣ ❲ ❚ ❥❥❥❥❥❥❥❥❥❥❥❥❥ ❲ ❍ ❚ ❲ ❲ ❚ ❍ ✈✈✈✈✈✈ ❲ ❚ ❲ ❍ ❲ ❚ ❲ ❚ ❲ ❍ ❲ ❚ ❲ ❚ ❍ ❲ ❚ ❲ ❲ ❚ ❍ ❲ ❚ ❲ ❲ ❚ ❲ • Theorem : There is no piecewise morphism ◮ Proj( C 3 ) → { 0 , 1 } 15 / 31
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