Free monotone transport without a trace Brent Nelson UCLA October - PowerPoint PPT Presentation

Preliminaries Setup Ω is cyclic and separating for Γ q ( H R , U t ) ′′ and hence the vector state ϕ ( · ) = � Ω , · Ω � U , q is a faithful, non-degenerate state ( free quasi-free state Throughout, M shall denote Γ 0 ( H R , U t ) ′′ = W ∗ ( X 1 , . . . , X N ), with X j := s 0 ( e j ). With respect to the vacuum vector state ϕ , the X j are centered semicircular random variables of variance 1, but aren’t free unless U t = id . Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 7 / 38

Preliminaries Setup Ω is cyclic and separating for Γ q ( H R , U t ) ′′ and hence the vector state ϕ ( · ) = � Ω , · Ω � U , q is a faithful, non-degenerate state ( free quasi-free state Throughout, M shall denote Γ 0 ( H R , U t ) ′′ = W ∗ ( X 1 , . . . , X N ), with X j := s 0 ( e j ). With respect to the vacuum vector state ϕ , the X j are centered semicircular random variables of variance 1, but aren’t free unless U t = id . Application of result: for small values of | q | , Γ q ( H R , U t ) ′′ is isomorphic to M . Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 7 / 38

Preliminaries Tomita-Takesaki theory Modular group: σ ϕ z ( X j ) = � N k =1 [ A iz ] jk X k for z ∈ C Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 8 / 38

Preliminaries Tomita-Takesaki theory Modular group: σ ϕ z ( X j ) = � N k =1 [ A iz ] jk X k for z ∈ C Using the vector notation X = ( X 1 , . . . , X N ) we have σ ϕ z ( X ) = A iz X . Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 8 / 38

Preliminaries Tomita-Takesaki theory Modular group: σ ϕ z ( X j ) = � N k =1 [ A iz ] jk X k for z ∈ C Using the vector notation X = ( X 1 , . . . , X N ) we have σ ϕ z ( X ) = A iz X . KMS condition: ϕ ( X j P ) = ϕ ( P σ − i ( X j )) = ϕ ( P [ AX ] j ) ϕ ( PX j ) = ϕ ( σ i ( X j ) P ) = ϕ ([ A − 1 X ] j P ) . Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 8 / 38

Preliminaries Banach algebras and norms P := C � X 1 , . . . , X N � ⊂ M . Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 9 / 38

Preliminaries Banach algebras and norms P := C � X 1 , . . . , X N � ⊂ M . Can write each P ∈ P as deg( P ) deg( P ) � � � P = c ( j ) X j = π n ( P ) , c ( j ) ∈ C n =0 n =0 | j | = n Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 9 / 38

Preliminaries Banach algebras and norms P := C � X 1 , . . . , X N � ⊂ M . Can write each P ∈ P as deg( P ) deg( P ) � � � P = c ( j ) X j = π n ( P ) , c ( j ) ∈ C n =0 n =0 | j | = n For R > 0 deg( P ) | c ( j ) | R n = � � � � P � R := � π n ( P ) � R , n =0 n | j | = n defines a Banach norm on P . Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 9 / 38

Preliminaries Banach algebras and norms P := C � X 1 , . . . , X N � ⊂ M . Can write each P ∈ P as deg( P ) deg( P ) � � � P = c ( j ) X j = π n ( P ) , c ( j ) ∈ C n =0 n =0 | j | = n For R > 0 deg( P ) | c ( j ) | R n = � � � � P � R := � π n ( P ) � R , n =0 n | j | = n defines a Banach norm on P . P ( R ) = P �·� R Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 9 / 38

Preliminaries Banach algebras and norms P := C � X 1 , . . . , X N � ⊂ M . Can write each P ∈ P as deg( P ) deg( P ) � � � P = c ( j ) X j = π n ( P ) , c ( j ) ∈ C n =0 n =0 | j | = n For R > 0 deg( P ) | c ( j ) | R n = � � � � P � R := � π n ( P ) � R , n =0 n | j | = n defines a Banach norm on P . P ( R ) = P �·� R If R ≥ 2 ≥ � X j � , then P ( R ) ⊂ M . Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 9 / 38

Preliminaries Banach algebras and norms P ϕ = { P ∈ P : σ i ( P ) = P } = M ϕ ∩ P . Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 10 / 38

Preliminaries Banach algebras and norms P ϕ = { P ∈ P : σ i ( P ) = P } = M ϕ ∩ P . Define ρ : P → P on monomials by ρ ( X j 1 · · · X j n ) = σ − i ( X j n ) X i 1 · · · X j n − 1 . Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 10 / 38

Preliminaries Banach algebras and norms P ϕ = { P ∈ P : σ i ( P ) = P } = M ϕ ∩ P . Define ρ : P → P on monomials by ρ ( X j 1 · · · X j n ) = σ − i ( X j n ) X i 1 · · · X j n − 1 . We call ρ k ( P ) for k ∈ Z a σ -cyclic rearrangement of P . Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 10 / 38

Preliminaries Banach algebras and norms P ϕ = { P ∈ P : σ i ( P ) = P } = M ϕ ∩ P . Define ρ : P → P on monomials by ρ ( X j 1 · · · X j n ) = σ − i ( X j n ) X i 1 · · · X j n − 1 . We call ρ k ( P ) for k ∈ Z a σ -cyclic rearrangement of P . Define deg( P ) � � � � ρ k n ( π n ( P )) � P � R ,σ = sup R , � � � k n ∈ Z n =0 is a Banach norm on P finite = { P ∈ P : � P � R ,σ < ∞} . Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 10 / 38

Preliminaries Banach algebras and norms P ϕ = { P ∈ P : σ i ( P ) = P } = M ϕ ∩ P . Define ρ : P → P on monomials by ρ ( X j 1 · · · X j n ) = σ − i ( X j n ) X i 1 · · · X j n − 1 . We call ρ k ( P ) for k ∈ Z a σ -cyclic rearrangement of P . Define deg( P ) � � � � ρ k n ( π n ( P )) � P � R ,σ = sup R , � � � k n ∈ Z n =0 is a Banach norm on P finite = { P ∈ P : � P � R ,σ < ∞} . P ϕ ⊂ P finite , in fact � P � R ,σ ≤ � A � deg( P ) − 1 � P � R . Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 10 / 38

Preliminaries Banach algebras and norms P ϕ = { P ∈ P : σ i ( P ) = P } = M ϕ ∩ P . Define ρ : P → P on monomials by ρ ( X j 1 · · · X j n ) = σ − i ( X j n ) X i 1 · · · X j n − 1 . We call ρ k ( P ) for k ∈ Z a σ -cyclic rearrangement of P . Define deg( P ) � � � � ρ k n ( π n ( P )) � P � R ,σ = sup R , � � � k n ∈ Z n =0 is a Banach norm on P finite = { P ∈ P : � P � R ,σ < ∞} . P ϕ ⊂ P finite , in fact � P � R ,σ ≤ � A � deg( P ) − 1 � P � R . P ( R ,σ ) = P finite �·� R ,σ Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 10 / 38

Preliminaries Banach algebras and norms We let P ( R ) and P ( R ,σ ) denote the elements of the respective ϕ ϕ algebras which are fixed by σ i . Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 11 / 38

Preliminaries Banach algebras and norms We let P ( R ) and P ( R ,σ ) denote the elements of the respective ϕ ϕ algebras which are fixed by σ i . = { P : P ( R ,σ ) : ρ ( P ) = P } be the σ -cyclically symmetric Let P ( R ,σ ) c . s . elements. Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 11 / 38

Preliminaries Banach algebras and norms We let P ( R ) and P ( R ,σ ) denote the elements of the respective ϕ ϕ algebras which are fixed by σ i . = { P : P ( R ,σ ) : ρ ( P ) = P } be the σ -cyclically symmetric Let P ( R ,σ ) c . s . elements. P ( R ) � N and P ( R ,σ ) � N we use the max-norm, which we still � � On denote � · � R and � · � R ,σ . Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 11 / 38

Preliminaries Differential operators Let δ j : P → P ⊗ P op be Voiculescu’s free difference quotients, defined by δ j ( X k ) = δ j = k 1 ⊗ 1 and the Leibniz rule. Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 12 / 38

Preliminaries Differential operators Let δ j : P → P ⊗ P op be Voiculescu’s free difference quotients, defined by δ j ( X k ) = δ j = k 1 ⊗ 1 and the Leibniz rule. Conventions on P ⊗ P op : Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 12 / 38

Preliminaries Differential operators Let δ j : P → P ⊗ P op be Voiculescu’s free difference quotients, defined by δ j ( X k ) = δ j = k 1 ⊗ 1 and the Leibniz rule. Conventions on P ⊗ P op : Suppress “ ◦ ” notation: a ⊗ b ◦ �→ a ⊗ b a ⊗ b # c ⊗ d = ( ac ) ⊗ ( db ) a ⊗ b # c = acb , m ( a ⊗ b ) = ab Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 12 / 38

Preliminaries Differential operators Let δ j : P → P ⊗ P op be Voiculescu’s free difference quotients, defined by δ j ( X k ) = δ j = k 1 ⊗ 1 and the Leibniz rule. Conventions on P ⊗ P op : Suppress “ ◦ ” notation: a ⊗ b ◦ �→ a ⊗ b a ⊗ b # c ⊗ d = ( ac ) ⊗ ( db ) a ⊗ b # c = acb , m ( a ⊗ b ) = ab ( a ⊗ b ) ∗ = a ∗ ⊗ b ∗ ( a ⊗ b ) † = b ∗ ⊗ a ∗ ( a ⊗ b ) ⋄ = b ⊗ a Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 12 / 38

Preliminaries Differential operators Let δ j : P → P ⊗ P op be Voiculescu’s free difference quotients, defined by δ j ( X k ) = δ j = k 1 ⊗ 1 and the Leibniz rule. Conventions on P ⊗ P op : Suppress “ ◦ ” notation: a ⊗ b ◦ �→ a ⊗ b a ⊗ b # c ⊗ d = ( ac ) ⊗ ( db ) a ⊗ b # c = acb , m ( a ⊗ b ) = ab ( a ⊗ b ) ∗ = a ∗ ⊗ b ∗ ( a ⊗ b ) † = b ∗ ⊗ a ∗ ( a ⊗ b ) ⋄ = b ⊗ a As a P − P bimodule: c · ( a ⊗ b ) = ( ca ) ⊗ b and ( a ⊗ b ) · c = a ⊗ ( bc ) Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 12 / 38

Preliminaries Differential operators For j , k ∈ { 1 , . . . , N } denote � 2 � α jk = = ϕ ( X k X j ) , 1 + A jk then α jk = α kj , α jj = 1, and | α jk | ≤ 1. Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 13 / 38

Preliminaries Differential operators For j , k ∈ { 1 , . . . , N } denote � 2 � α jk = = ϕ ( X k X j ) , 1 + A jk then α jk = α kj , α jj = 1, and | α jk | ≤ 1. For each j define σ -difference quotient ∂ j = � N k =1 α kj δ k Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 13 / 38

Preliminaries Differential operators For j , k ∈ { 1 , . . . , N } denote � 2 � α jk = = ϕ ( X k X j ) , 1 + A jk then α jk = α kj , α jj = 1, and | α jk | ≤ 1. For each j define σ -difference quotient ∂ j = � N k =1 α kj δ k We consider this derivation because ϕ ( X j P ) = ϕ ⊗ ϕ op ( ∂ j ( P )) for P ∈ P . Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 13 / 38

Preliminaries Differential operators For j , k ∈ { 1 , . . . , N } denote � 2 � α jk = = ϕ ( X k X j ) , 1 + A jk then α jk = α kj , α jj = 1, and | α jk | ≤ 1. For each j define σ -difference quotient ∂ j = � N k =1 α kj δ k We consider this derivation because ϕ ( X j P ) = ϕ ⊗ ϕ op ( ∂ j ( P )) for P ∈ P . ∂ j so that ∂ j ( P ) † = ¯ Define another derivation ¯ ∂ j ( P ∗ ). Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 13 / 38

Preliminaries Differential operators For j , k ∈ { 1 , . . . , N } denote � 2 � α jk = = ϕ ( X k X j ) , 1 + A jk then α jk = α kj , α jj = 1, and | α jk | ≤ 1. For each j define σ -difference quotient ∂ j = � N k =1 α kj δ k We consider this derivation because ϕ ( X j P ) = ϕ ⊗ ϕ op ( ∂ j ( P )) for P ∈ P . ∂ j so that ∂ j ( P ) † = ¯ Define another derivation ¯ ∂ j ( P ∗ ). The modular group interacts with ∂ j as follows: ( σ i ⊗ σ i ) ◦ ∂ j ◦ σ − i = ¯ ∂ j Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 13 / 38

Preliminaries Differential operators For P = ( P 1 , . . . , P N ) ∈ P N define J P , J σ P ∈ M N ( P ⊗ P op ) by [ J P ] jk = δ k P j [ J σ P ] jk = ∂ k P j Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 14 / 38

Preliminaries Differential operators For P = ( P 1 , . . . , P N ) ∈ P N define J P , J σ P ∈ M N ( P ⊗ P op ) by [ J P ] jk = δ k P j [ J σ P ] jk = ∂ k P j → M N ( P ⊗ P op ) in the obvious way. M N ( C ) ֒ Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 14 / 38

Preliminaries Differential operators For P = ( P 1 , . . . , P N ) ∈ P N define J P , J σ P ∈ M N ( P ⊗ P op ) by [ J P ] jk = δ k P j [ J σ P ] jk = ∂ k P j → M N ( P ⊗ P op ) in the obvious way. M N ( C ) ֒ Examples: [ J X ] jk = δ k X j = δ k = j 1 ⊗ 1 = [1] jk � 2 � [ J σ X ] jk = ∂ k X j = α jk 1 ⊗ 1 = 1 + A jk Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 14 / 38

Preliminaries Differential operators For P = ( P 1 , . . . , P N ) ∈ P N define J P , J σ P ∈ M N ( P ⊗ P op ) by [ J P ] jk = δ k P j [ J σ P ] jk = ∂ k P j → M N ( P ⊗ P op ) in the obvious way. M N ( C ) ֒ Examples: [ J X ] jk = δ k X j = δ k = j 1 ⊗ 1 = [1] jk � 2 � [ J σ X ] jk = ∂ k X j = α jk 1 ⊗ 1 = 1 + A jk A simple computation reveals J P = J σ P # J σ X − 1 for all P ∈ ( P ( R ) ) N . Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 14 / 38

Preliminaries Differential operators For each j we define the j-th σ -cyclic derivative D j : P → P by n � D j ( X k 1 · · · X k n ) = α jk l σ − i ( X k l +1 · · · X k n ) X k 1 · · · X k l − 1 , l =1 or D j = m ◦ ⋄ ◦ (1 ⊗ σ − i ) ◦ ¯ ∂ j . Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 15 / 38

Preliminaries Differential operators For each j we define the j-th σ -cyclic derivative D j : P → P by n � D j ( X k 1 · · · X k n ) = α jk l σ − i ( X k l +1 · · · X k n ) X k 1 · · · X k l − 1 , l =1 or D j = m ◦ ⋄ ◦ (1 ⊗ σ − i ) ◦ ¯ ∂ j . We define the σ -cyclic gradient by D P = ( D 1 P , . . . , D N P ) ∈ P N for P ∈ P . Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 15 / 38

Preliminaries Differential operators For each j we define the j-th σ -cyclic derivative D j : P → P by n � D j ( X k 1 · · · X k n ) = α jk l σ − i ( X k l +1 · · · X k n ) X k 1 · · · X k l − 1 , l =1 or D j = m ◦ ⋄ ◦ (1 ⊗ σ − i ) ◦ ¯ ∂ j . We define the σ -cyclic gradient by D P = ( D 1 P , . . . , D N P ) ∈ P N for P ∈ P . Example: N � 1 + A � V 0 = 1 X k X j ∈ P ( R ,σ ) � c . s . 2 2 jk j , k =1 then D V 0 = ( X 1 , . . . , X N ) = X . Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 15 / 38

Preliminaries Differential operators For each j we define the j-th σ -cyclic derivative D j : P → P by n � D j ( X k 1 · · · X k n ) = α jk l σ − i ( X k l +1 · · · X k n ) X k 1 · · · X k l − 1 , l =1 or D j = m ◦ ⋄ ◦ (1 ⊗ σ − i ) ◦ ¯ ∂ j . We define the σ -cyclic gradient by D P = ( D 1 P , . . . , D N P ) ∈ P N for P ∈ P . Example: N � 1 + A � V 0 = 1 X k X j ∈ P ( R ,σ ) � c . s . 2 2 jk j , k =1 then D V 0 = ( X 1 , . . . , X N ) = X . D j so that ( D j P ) ∗ = ¯ Can also define ¯ D j ( P ∗ ). Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 15 / 38

Preliminaries Schwinger-Dyson equation Given V ∈ P ( R ,σ ) c . s . , we say that a state ψ on W ∗ ( X 1 , . . . , X N ) satisfies the Schwinger-Dyson equation with potential V if ψ ( D V # P ) = ψ ⊗ ψ op ⊗ Tr( J σ P ) ∀ P ∈ P ( R ) , in which case we call ψ the free Gibbs state with potential V , and may denote it ϕ V . Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 16 / 38

Preliminaries Schwinger-Dyson equation Given V ∈ P ( R ,σ ) c . s . , we say that a state ψ on W ∗ ( X 1 , . . . , X N ) satisfies the Schwinger-Dyson equation with potential V if ψ ( D V # P ) = ψ ⊗ ψ op ⊗ Tr( J σ P ) ∀ P ∈ P ( R ) , in which case we call ψ the free Gibbs state with potential V , and may denote it ϕ V . The state ϕ V is unique provided � V − V 0 � R ,σ is small enough. Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 16 / 38

Preliminaries Schwinger-Dyson equation Given V ∈ P ( R ,σ ) c . s . , we say that a state ψ on W ∗ ( X 1 , . . . , X N ) satisfies the Schwinger-Dyson equation with potential V if ψ ( D V # P ) = ψ ⊗ ψ op ⊗ Tr( J σ P ) ∀ P ∈ P ( R ) , in which case we call ψ the free Gibbs state with potential V , and may denote it ϕ V . The state ϕ V is unique provided � V − V 0 � R ,σ is small enough. The vacuum vector state ϕ = ϕ V 0 . Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 16 / 38

Preliminaries Schwinger-Dyson equation Given V ∈ P ( R ,σ ) c . s . , we say that a state ψ on W ∗ ( X 1 , . . . , X N ) satisfies the Schwinger-Dyson equation with potential V if ψ ( D V # P ) = ψ ⊗ ψ op ⊗ Tr( J σ P ) ∀ P ∈ P ( R ) , in which case we call ψ the free Gibbs state with potential V , and may denote it ϕ V . The state ϕ V is unique provided � V − V 0 � R ,σ is small enough. The vacuum vector state ϕ = ϕ V 0 . Consequently, X = J ∗ σ (1), where 1 ∈ M N ( P ⊗ P op ) is the identity matrix. Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 16 / 38

Preliminaries Schwinger-Dyson equation Idea is to suppose the law of Z = ( Z 1 , . . . , Z N ) ⊂ ( L , ψ ) is the free Gibbs state with potential V = V 0 + W : ψ Z = ϕ V . Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 17 / 38

Preliminaries Schwinger-Dyson equation Idea is to suppose the law of Z = ( Z 1 , . . . , Z N ) ⊂ ( L , ψ ) is the free Gibbs state with potential V = V 0 + W : ψ Z = ϕ V . By exploiting the Schwinger-Dyson equation, we will construct Y = ( Y 1 , . . . , Y N ) ⊂ ( M , ϕ ) of the form Y j = X j + f j whose law induced by ϕ is also the free Gibbs state with potential V . Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 17 / 38

Preliminaries Schwinger-Dyson equation Idea is to suppose the law of Z = ( Z 1 , . . . , Z N ) ⊂ ( L , ψ ) is the free Gibbs state with potential V = V 0 + W : ψ Z = ϕ V . By exploiting the Schwinger-Dyson equation, we will construct Y = ( Y 1 , . . . , Y N ) ⊂ ( M , ϕ ) of the form Y j = X j + f j whose law induced by ϕ is also the free Gibbs state with potential V . Provided � W � R ,σ is small enough, the free Gibbs state with potential V 0 + W will be unique and therefore we will have transport from ϕ X to ψ Z . Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 17 / 38

Construction of transport Equivalent forms of Schwinger-Dyson Suppose Y = ( Y 1 , . . . , Y N ) with Y j = X j + f j and f j ∈ P ( R ) , assume assume that ϕ Y satisfies the Schwinger-Dyson equation with potential V = V 0 + W . Then ( J σ ) ∗ Y (1) = D Y ( V 0 ( Y ) + W ( Y )) = Y + ( D W )( Y ) (1) = X + f + ( D W )( X + f ) Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 18 / 38

Construction of transport Equivalent forms of Schwinger-Dyson Suppose Y = ( Y 1 , . . . , Y N ) with Y j = X j + f j and f j ∈ P ( R ) , assume assume that ϕ Y satisfies the Schwinger-Dyson equation with potential V = V 0 + W . Then ( J σ ) ∗ Y (1) = D Y ( V 0 ( Y ) + W ( Y )) = Y + ( D W )( Y ) (1) = X + f + ( D W )( X + f ) Need to write the left-hand side in terms of X . Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 18 / 38

Construction of transport Equivalent forms of Schwinger-Dyson Using a change of variables argument, the Schwinger-Dyson equation (1) is equivalent to � 1 � J ∗ σ ◦ (1 ⊗ σ i ) = X + f + ( D W )( X + f ) , (2) 1 + B where B = J σ f # J σ X − 1 . Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 19 / 38

Construction of transport Equivalent forms of Schwinger-Dyson Using a change of variables argument, the Schwinger-Dyson equation (1) is equivalent to � 1 � J ∗ σ ◦ (1 ⊗ σ i ) = X + f + ( D W )( X + f ) , (2) 1 + B where B = J σ f # J σ X − 1 . x 2 1 x x Using identities 1+ x = 1 − 1+ x and 1+ x = x − 1+ x and multiplying by 1 + B , (2) becomes − J ∗ σ ◦ (1 ⊗ σ i )( B ) − f � B � = D ( W ( X + f )) + B # f + B # J ∗ σ ◦ (1 ⊗ σ i ) (3) 1 + B � B 2 � − J ∗ σ ◦ (1 ⊗ σ i ) , 1 + B Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 19 / 38

Construction of transport Equivalent forms of Schwinger-Dyson Lemma 2.1 Let g = g ∗ ∈ P ( R ,σ ) and let f = D g. Then for any m ≥ − 1 we have: ϕ B # J ∗ σ ◦ (1 ⊗ σ i )( B m +1 ) − J ∗ σ ◦ (1 ⊗ σ i )( B m +2 ) (4) 1 m + 2 D [( ϕ ⊗ 1) ◦ Tr A − 1 + (1 ⊗ ϕ ) ◦ Tr A ] ( B m +2 ) = Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 20 / 38

Construction of transport Equivalent forms of Schwinger-Dyson Lemma 2.1 Let g = g ∗ ∈ P ( R ,σ ) and let f = D g. Then for any m ≥ − 1 we have: ϕ B # J ∗ σ ◦ (1 ⊗ σ i )( B m +1 ) − J ∗ σ ◦ (1 ⊗ σ i )( B m +2 ) (4) 1 m + 2 D [( ϕ ⊗ 1) ◦ Tr A − 1 + (1 ⊗ ϕ ) ◦ Tr A ] ( B m +2 ) = Proof. We prove the equivalence weakly by taking inner products against P ∈ ( P ( R ) ) N . Denote the left-hand side by E L and the right-hand side by E R . Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 20 / 38

Construction of transport Equivalent forms of Schwinger-Dyson Proof of Lemma 2.1 (conti.) � P , B # J ∗ σ ◦ (1 ⊗ σ i )( B m +1 ) � ϕ Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 21 / 38

Construction of transport Equivalent forms of Schwinger-Dyson Proof of Lemma 2.1 (conti.) � P , B # J ∗ σ ◦ (1 ⊗ σ i )( B m +1 ) � ϕ N � P ∗ J ∗ σ ◦ (1 ⊗ σ i )( B m +1 ) � � � � = ϕ j · B jk # k j , k =1 Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 21 / 38

Construction of transport Equivalent forms of Schwinger-Dyson Proof of Lemma 2.1 (conti.) � P , B # J ∗ σ ◦ (1 ⊗ σ i )( B m +1 ) � ϕ N � P ∗ J ∗ σ ◦ (1 ⊗ σ i )( B m +1 ) � � � � = ϕ j · B jk # k j , k =1 N � ( σ i ⊗ 1)( B ⋄ jk )# P ∗ J ∗ σ ◦ (1 ⊗ σ i )( B m +1 ) � � � � = ϕ j · k j , k =1 Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 21 / 38

Construction of transport Equivalent forms of Schwinger-Dyson Proof of Lemma 2.1 (conti.) � P , B # J ∗ σ ◦ (1 ⊗ σ i )( B m +1 ) � ϕ N � P ∗ J ∗ σ ◦ (1 ⊗ σ i )( B m +1 ) � � � � = ϕ j · B jk # k j , k =1 N � ( σ i ⊗ 1)( B ⋄ jk )# P ∗ J ∗ σ ◦ (1 ⊗ σ i )( B m +1 ) � � � � = ϕ j · k j , k =1 (1 ⊗ σ − i )( B ∗ )# P , J ∗ σ ◦ (1 ⊗ σ i )( B m +1 ) � � = ϕ Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 21 / 38

Construction of transport Equivalent forms of Schwinger-Dyson Proof of Lemma 2.1 (conti.) � P , B # J ∗ σ ◦ (1 ⊗ σ i )( B m +1 ) � ϕ N � P ∗ J ∗ σ ◦ (1 ⊗ σ i )( B m +1 ) � � � � = ϕ j · B jk # k j , k =1 N � ( σ i ⊗ 1)( B ⋄ jk )# P ∗ J ∗ σ ◦ (1 ⊗ σ i )( B m +1 ) � � � � = ϕ j · k j , k =1 (1 ⊗ σ − i )( B ∗ )# P , J ∗ σ ◦ (1 ⊗ σ i )( B m +1 ) � � = ϕ [ J σ X − 1 #ˆ σ i ( J σ f )]# P , J ∗ σ ◦ (1 ⊗ σ i )( B m +1 ) � � = ϕ where ˆ σ i = σ i ⊗ σ − i . Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 21 / 38

Construction of transport Equivalent forms of Schwinger-Dyson Proof of Lemma 2.1 (conti.) Hence if φ = ϕ ⊗ ϕ op ⊗ Tr then J σ X − 1 # J σ { ˆ σ i ( J σ f )# P } , (1 ⊗ σ i )( B m +1 ) � � � P , E L � ϕ = φ J σ P , (1 ⊗ σ i )( B m +2 ) � � − φ . Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 22 / 38

Construction of transport Equivalent forms of Schwinger-Dyson Proof of Lemma 2.1 (conti.) Hence if φ = ϕ ⊗ ϕ op ⊗ Tr then J σ X − 1 # J σ { ˆ σ i ( J σ f )# P } , (1 ⊗ σ i )( B m +1 ) � � � P , E L � ϕ = φ J σ P , (1 ⊗ σ i )( B m +2 ) � � − φ . The “product rule” simplifies the right-hand side to simplify to � � Q P , J σ X − 1 #(1 ⊗ σ i )( B m +1 ) � P , E L � ϕ = φ , Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 22 / 38

Construction of transport Equivalent forms of Schwinger-Dyson Proof of Lemma 2.1 (conti.) Hence if φ = ϕ ⊗ ϕ op ⊗ Tr then J σ X − 1 # J σ { ˆ σ i ( J σ f )# P } , (1 ⊗ σ i )( B m +1 ) � � � P , E L � ϕ = φ J σ P , (1 ⊗ σ i )( B m +2 ) � � − φ . The “product rule” simplifies the right-hand side to simplify to � � Q P , J σ X − 1 #(1 ⊗ σ i )( B m +1 ) � P , E L � ϕ = φ , where, if a ⊗ b ⊗ c # 1 ξ = ( a ξ b ) ⊗ c and a ⊗ b ⊗ c # 2 ξ = a ⊗ ( b ξ c ), then Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 22 / 38

Construction of transport Equivalent forms of Schwinger-Dyson Proof of Lemma 2.1 (conti.) Hence if φ = ϕ ⊗ ϕ op ⊗ Tr then J σ X − 1 # J σ { ˆ σ i ( J σ f )# P } , (1 ⊗ σ i )( B m +1 ) � � � P , E L � ϕ = φ J σ P , (1 ⊗ σ i )( B m +2 ) � � − φ . The “product rule” simplifies the right-hand side to simplify to � � Q P , J σ X − 1 #(1 ⊗ σ i )( B m +1 ) � P , E L � ϕ = φ , where, if a ⊗ b ⊗ c # 1 ξ = ( a ξ b ) ⊗ c and a ⊗ b ⊗ c # 2 ξ = a ⊗ ( b ξ c ), then N [ Q P ] jk = � ( ∂ k ⊗ 1) ◦ ˆ σ i ◦ ∂ l ( f j )# 2 P l + (1 ⊗ ∂ k ) ◦ ˆ σ i ◦ ∂ l ( f j )# 1 P l l =1 Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 22 / 38

Construction of transport Equivalent forms of Schwinger-Dyson Proof of Lemma 2.1 (conti.) So we have � E L , P � ϕ = φ ( Q P # J σ X − 1 # B m +1 ) Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 23 / 38

Construction of transport Equivalent forms of Schwinger-Dyson Proof of Lemma 2.1 (conti.) Next for u = 1 , . . . , m + 2 let R u be the matrix will all zero entries except [ R u ] i u j u = a u ⊗ b u . Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 24 / 38

Construction of transport Equivalent forms of Schwinger-Dyson Proof of Lemma 2.1 (conti.) Next for u = 1 , . . . , m + 2 let R u be the matrix will all zero entries except � m +2 [ R u ] i u j u = a u ⊗ b u . Let C = [ A − 1 ] j m +2 i 1 u =1 δ j u = i u +1 and consider Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 24 / 38

Construction of transport Equivalent forms of Schwinger-Dyson Proof of Lemma 2.1 (conti.) Next for u = 1 , . . . , m + 2 let R u be the matrix will all zero entries except � m +2 [ R u ] i u j u = a u ⊗ b u . Let C = [ A − 1 ] j m +2 i 1 u =1 δ j u = i u +1 and consider � ϕ ( ¯ D k ( ϕ ⊗ 1)Tr A − 1 ( R 1 · · · R m +2 ) P k ) k Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 24 / 38

Construction of transport Equivalent forms of Schwinger-Dyson Proof of Lemma 2.1 (conti.) Next for u = 1 , . . . , m + 2 let R u be the matrix will all zero entries except � m +2 [ R u ] i u j u = a u ⊗ b u . Let C = [ A − 1 ] j m +2 i 1 u =1 δ j u = i u +1 and consider � ϕ ( ¯ D k ( ϕ ⊗ 1)Tr A − 1 ( R 1 · · · R m +2 ) P k ) k � C ϕ ( a 1 · · · a m +2 ) ϕ ( ¯ = D k ( b m +2 · · · b 1 ) P k ) k Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 24 / 38

Construction of transport Equivalent forms of Schwinger-Dyson Proof of Lemma 2.1 (conti.) Next for u = 1 , . . . , m + 2 let R u be the matrix will all zero entries except � m +2 [ R u ] i u j u = a u ⊗ b u . Let C = [ A − 1 ] j m +2 i 1 u =1 δ j u = i u +1 and consider � ϕ ( ¯ D k ( ϕ ⊗ 1)Tr A − 1 ( R 1 · · · R m +2 ) P k ) k � C ϕ ( a 1 · · · a m +2 ) ϕ ( ¯ = D k ( b m +2 · · · b 1 ) P k ) k � = C ϕ ( a 1 · · · a m +2 ) ϕ (ˆ σ i ◦ ∂ k ( b m +2 · · · b 1 )# P k ) k Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 24 / 38

Construction of transport Equivalent forms of Schwinger-Dyson Proof of Lemma 2.1 (conti.) Next for u = 1 , . . . , m + 2 let R u be the matrix will all zero entries except � m +2 [ R u ] i u j u = a u ⊗ b u . Let C = [ A − 1 ] j m +2 i 1 u =1 δ j u = i u +1 and consider � ϕ ( ¯ D k ( ϕ ⊗ 1)Tr A − 1 ( R 1 · · · R m +2 ) P k ) k � C ϕ ( a 1 · · · a m +2 ) ϕ ( ¯ = D k ( b m +2 · · · b 1 ) P k ) k � = C ϕ ( a 1 · · · a m +2 ) ϕ (ˆ σ i ◦ ∂ k ( b m +2 · · · b 1 )# P k ) k � = C ϕ ( σ i ( a u · · · a m +2 ) a 1 · · · a u − 1 ) k , u × ϕ ( b u − 1 · · · b 1 σ i ( b m +2 · · · b u +1 ) · ˆ σ i ◦ ∂ k ( b u )# P k ) Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 24 / 38

Construction of transport Equivalent forms of Schwinger-Dyson Proof of Lemma 2.1 (conti.) Next for u = 1 , . . . , m + 2 let R u be the matrix will all zero entries except � m +2 [ R u ] i u j u = a u ⊗ b u . Let C = [ A − 1 ] j m +2 i 1 u =1 δ j u = i u +1 and consider � ϕ ( ¯ D k ( ϕ ⊗ 1)Tr A − 1 ( R 1 · · · R m +2 ) P k ) k � C ϕ ( a 1 · · · a m +2 ) ϕ ( ¯ = D k ( b m +2 · · · b 1 ) P k ) k � = C ϕ ( a 1 · · · a m +2 ) ϕ (ˆ σ i ◦ ∂ k ( b m +2 · · · b 1 )# P k ) k � = C ϕ ( σ i ( a u · · · a m +2 ) a 1 · · · a u − 1 ) k , u × ϕ ( b u − 1 · · · b 1 σ i ( b m +2 · · · b u +1 ) · ˆ σ i ◦ ∂ k ( b u )# P k ) � φ (∆ (1 , P ) ( R u )( σ i ⊗ σ i )( R u +1 · · · R m +2 ) A − 1 R 1 · · · R u − 1 ) = u Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 24 / 38

Construction of transport Equivalent forms of Schwinger-Dyson Proof of Lemma 2.1 (conti.) Where for an arbitrary matrix O � [∆ (1 , P ) ( O )] jk = σ i ⊗ (ˆ σ i ◦ ∂ l )([ O ] jk )# 2 P l . l Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 25 / 38

Construction of transport Equivalent forms of Schwinger-Dyson Proof of Lemma 2.1 (conti.) Where for an arbitrary matrix O � [∆ (1 , P ) ( O )] jk = σ i ⊗ (ˆ σ i ◦ ∂ l )([ O ] jk )# 2 P l . l Replacing R u with B for each u and using ( σ i ⊗ σ i )( B ) A − 1 = A − 1 B turns the previous equation into Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 25 / 38

Construction of transport Equivalent forms of Schwinger-Dyson Proof of Lemma 2.1 (conti.) Where for an arbitrary matrix O � [∆ (1 , P ) ( O )] jk = σ i ⊗ (ˆ σ i ◦ ∂ l )([ O ] jk )# 2 P l . l Replacing R u with B for each u and using ( σ i ⊗ σ i )( B ) A − 1 = A − 1 B turns the previous equation into � ϕ ( ¯ D k ( ϕ ⊗ 1)Tr A − 1 ( B m +2 ) P k ) k � φ (∆ (1 , P ) ( B )( σ i ⊗ σ i )( B m +2 − u ) A − 1 B u − 1 ) = u Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 25 / 38

Construction of transport Equivalent forms of Schwinger-Dyson Proof of Lemma 2.1 (conti.) Where for an arbitrary matrix O � [∆ (1 , P ) ( O )] jk = σ i ⊗ (ˆ σ i ◦ ∂ l )([ O ] jk )# 2 P l . l Replacing R u with B for each u and using ( σ i ⊗ σ i )( B ) A − 1 = A − 1 B turns the previous equation into � ϕ ( ¯ D k ( ϕ ⊗ 1)Tr A − 1 ( B m +2 ) P k ) k � φ (∆ (1 , P ) ( B )( σ i ⊗ σ i )( B m +2 − u ) A − 1 B u − 1 ) = u = ( m + 2) φ (∆ (1 , P ) ( B ) A − 1 B m +1 ) Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 25 / 38

Construction of transport Equivalent forms of Schwinger-Dyson Proof of Lemma 2.1 (conti.) Where for an arbitrary matrix O � [∆ (1 , P ) ( O )] jk = σ i ⊗ (ˆ σ i ◦ ∂ l )([ O ] jk )# 2 P l . l Replacing R u with B for each u and using ( σ i ⊗ σ i )( B ) A − 1 = A − 1 B turns the previous equation into � D ( ϕ ⊗ 1) Tr A − 1 ( B m +2 ) , P � ϕ � φ (∆ (1 , P ) ( B )( σ i ⊗ σ i )( B m +2 − u ) A − 1 B u − 1 ) = u = ( m + 2) φ (∆ (1 , P ) ( B ) A − 1 B m +1 ) Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 25 / 38

Construction of transport Equivalent forms of Schwinger-Dyson Proof of Lemma 2.1 (conti.) Similarly, D (1 ⊗ ϕ )Tr A ( B m +2 ) , P ϕ = ( m + 2) φ (∆ (2 , P ) ( B ) AB m +1 ) , � � where � [∆ (2 , P ) ( O )] jk = (ˆ σ i ◦ ∂ l ) ⊗ σ − i ([ O ] jk )# 1 P l . l Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 26 / 38

Construction of transport Equivalent forms of Schwinger-Dyson Proof of Lemma 2.1 (conti.) Similarly, D (1 ⊗ ϕ )Tr A ( B m +2 ) , P ϕ = ( m + 2) φ (∆ (2 , P ) ( B ) AB m +1 ) , � � where � [∆ (2 , P ) ( O )] jk = (ˆ σ i ◦ ∂ l ) ⊗ σ − i ([ O ] jk )# 1 P l . l To finish the proof we simply verify that Q P # J σ X − 1 = ∆ (1 , P ) ( B ) A − 1 + ∆ (2 , P ) ( B ) A , which follows from their definitions after decomposing the various derivations as linear combinations of the free difference quotients δ k . Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 26 / 38

Construction of transport Equivalent forms of Schwinger-Dyson Define N ( X i ) = | i | X i Σ( X i ) = 1 | i | X i Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 27 / 38

Construction of transport Equivalent forms of Schwinger-Dyson Recall f = D g , and B = J σ f # J σ X − 1 = J f . Set Q ( g ) = [(1 ⊗ ϕ ) ◦ Tr A + ( ϕ ⊗ 1) ◦ Tr A − 1 ]( B − log(1 + B )) , Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 28 / 38

Construction of transport Equivalent forms of Schwinger-Dyson Recall f = D g , and B = J σ f # J σ X − 1 = J f . Set Q ( g ) = [(1 ⊗ ϕ ) ◦ Tr A + ( ϕ ⊗ 1) ◦ Tr A − 1 ]( B − log(1 + B )) , Then by comparing power series the previous lemma implies � B 2 � B � � D Q ( g ) = B # J ∗ − J ∗ σ ◦ (1 ⊗ σ ) σ ◦ (1 ⊗ σ i ) . 1 + B 1 + B Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 28 / 38

Construction of transport Equivalent forms of Schwinger-Dyson Recall f = D g , and B = J σ f # J σ X − 1 = J f . Set Q ( g ) = [(1 ⊗ ϕ ) ◦ Tr A + ( ϕ ⊗ 1) ◦ Tr A − 1 ]( B − log(1 + B )) , Then by comparing power series the previous lemma implies � B 2 � B � � D Q ( g ) = B # J ∗ − J ∗ σ ◦ (1 ⊗ σ ) σ ◦ (1 ⊗ σ i ) . 1 + B 1 + B Lemma 2.2 Assume f = D g for g = g ∗ ∈ P ( R ,σ ) and � J D g � R ⊗ π R < 1 . Then ϕ equation (3) is equivalent to D { [( ϕ ⊗ 1) ◦ Tr A − 1 + (1 ⊗ ϕ ) ◦ Tr A ]( J D g ) − N g } (5) = D ( W ( X + D g )) + D Q ( g ) + ( J D g )# D g Brent Nelson (UCLA) Free monotone transport without a trace October 30, 2013 28 / 38