WHY IS THE WEYL QUANTIZATION THE BEST? JAN DEREZI NSKI Dept. of - PowerPoint PPT Presentation

WHY IS THE WEYL QUANTIZATION THE BEST? JAN DEREZI´ NSKI Dept. of Math. Methods in Phys., Faculty of Physics, University of Warsaw

Basic classical mechanics – phase space R 2 d with generic variables ( x i , p j ) . Basic quantum mechanics – Hilbert space L 2 ( R d ) with self-adjoint ∂ x i , ˆ p j := � operators ˆ ∂x j , where � is a small parameter. i A linear transformation which to a complex function b on R 2 d associates an operator Op • ( b ) on L 2 ( R d ) is often called a quantization of the symbol b .

Desirable properties: (1) Op • (1) = 1 l , Op • ( x i ) = ˆ x i , Op • ( p j ) = ˆ p j ; x ) = Op • � i i x ) Op • ( b )e � ( − y ˆ p + w ˆ � ( y ˆ p − w ˆ � (2) e b ( · − y, · − w ) . Op • ( b )Op • ( c ) + Op • ( c )Op • ( b ) ≈ Op • ( bc ) ; (3) 1 � � 2 Op • ( b ) , Op • ( c )] ≈ i � Op • ( { b, c } ) ; � (4) [

Let us strengthen the desirable property (1) to Op • ( f ( x )) = f (ˆ x ) , Op • ( g ( p )) = g (ˆ p ) . The so-called x, p -quantization is determined by the additional con- dition Op x,p ( f ( x ) g ( p )) = f (ˆ x ) g (ˆ p ) . It is defined by � � i( x − y ) p Op x,p ( b )Ψ ( x ) = (2 π � ) − d � � d p d yb ( x, p )e Ψ( y ) . �

In terms of its distributional kernel one can write � i( x − y ) p Op x,p ( b )( x, y ) = (2 π � ) − d d pb ( x, p )e � We also have the closely related p, x -quantization, � i( x − y ) p Op p,x ( b )( x, y ) = (2 π � ) − d d pb ( y, p )e � We have Op x,p ( b ) ∗ = Op p,x ( b ) .

The Weyl quantization (or the Weyl-Wigner-Moyal quantization) is a compromise between the two above quantizations: � � x + y i( x − y ) p � Op( b )( x, y ) = (2 π � ) − d d pb , p e . � 2 If Op( b ) = B, the function b is often called the Wigner function or the Weyl symbol of the operator B : � x + z 2 , x − z � � e − i zp � d z. b ( x, p ) = B 2 We have Op( b ) ∗ = Op( b ) .

Hermann Weyl Eugene Wigner

Fix a normalized vector Ψ ∈ L 2 ( R d ) . Define y, w ∈ R d ⊕ R d , i � ( − y ˆ p + w ˆ x ) Ψ , Ψ ( y,w ) := e sometimes called the family of coherent states associated with Ψ . We have a continuous decomposition of identity � (2 π � ) − d | Ψ ( y,w ) )(Ψ ( y,w ) | d y d w = 1 l .

Let b be a function on te phase space. We define its contravariant quantization by � Op ct ( b ) := (2 π � ) − d | Ψ ( x,p ) )(Ψ ( x,p ) | b ( x, p )d x d p. If B = Op ct ( b ) , then b is called the contravariant symbol of B . Let b ≥ 0 . Then Op ct ( b ) ≥ 0 .

Let B ∈ B ( H ) . Then we define its covariant symbol by � � b ( x, p ) := Ψ ( x,p ) | B Ψ ( x,p ) . B is then called the covariant quantization of b and is denoted by Op cv ( b ) = B. Let Op cv ( b ) ≥ 0 . Then b ≥ 0 .

Introduce complex coordinates a i = (2 � ) − 1 / 2 ( x i + i p i ) , a ∗ i = (2 � ) − 1 / 2 ( x i − i p i ) . and operators a i = (2 � ) − 1 / 2 (ˆ ˆ x i + iˆ p i ) , a ∗ i = (2 � ) − 1 / 2 (ˆ x i − iˆ ˆ p i ) .

Consider a polynomial function on the phase space: � w α,β x α p β . w ( x, p ) = α,β It is easy to describe the x, p and p, x quantizations of w in terms of ordering the positions and momenta: � Op x,p ( w ) = x α ˆ p β , w α,β ˆ α,β � Op p,x ( w ) = p β ˆ x α . w α,β ˆ α,β The Weyl quantization amounts to the full symmetrization of ˆ x i and ˆ p j .

We can also rewrite the polynomial in terms of a i , a ∗ i . Thus we obtain w γ,δ a ∗ γ a δ =: ˜ � w ( a ∗ , a ) . w ( x, p ) = ˜ γ,δ Then we can introduce the Wick quantization Op a ∗ ,a ( w ) = � a ∗ γ ˆ a δ w γ,δ ˆ ˜ γ,δ and the anti-Wick quantization Op a,a ∗ ( w ) = � a δ ˆ a ∗ γ . w γ,δ ˆ ˜ γ,δ

4 e − 1 Consider the Gaussian vector Ω( x ) = ( π � ) − d 2 � x 2 . It is killed by the annihilation operators: ˆ a i Ω = 0 . Theorem (1) The Wick quantization coincides with the covariant quantiza- tion for Gaussian coherent states. (2) The anti-Wick quantization coincides with the contravariant quantization for Gaussian coherent states.

For Gaussian states one uses several alternative names of the co- variant and contravariant symbol of an operator. For contravariant symbol: anti-Wick symbol, Glauber-Sudarshan function, P-function. For covariant symbol: Wick symbol, Husimi or Husimi-Kano func- tion, Q-function. We will use the terms Wick/anti-Wick quantization/symbol.

Gian-Carlo Wick Roy J. Glauber George Sudarshan

5 most natural quantizations in the Berezin diagram: anti-Wick quantization  � e − � 4 ( D 2 x + D 2 p )  i � i � 2 D x · D p 2 D x · D p p, x e Weyl-Wigner e x, p − → − → quantization quantization quantization  � e − � 4 ( D 2 x + D 2 p )  Wick quantization

The x, p - and p, x quantizations are invariant wrt the group GL ( R d ) of linear transformations of the configuration space. The Wick and anti-Wick quantizations are invariant wrt the uni- tary group U ( C d ) , generated by all harmonic oscillators whose ground state is the given Gaussian state. The Weyl-Wigner quantization is invariant with respect to the symplectic group Sp ( R 2 d ) .

For Op( b )Op( c ) = Op( d ) we have � i 2 � ( D p 1 D x 2 − D x 1 D p 2 ) b ( x 1 , p 1 ) c ( x 2 , p 2 ) d ( x, p ) = e , � � x := x 1 = x 2 , p := p 1 = p 2 . where D y := 1 i ∂ y . Often one denotes d by b ∗ c . It is called the star or the Moyal product of b and c .

Consequences: 1 = Op( bc ) + O ( � 2 ) , � � Op( b )Op( c ) + Op( c )Op( b ) 2 [Op( b ) , Op( c )] = i � Op( { b, c } ) + O ( � 3 ) , Op( b )Op( c ) = O ( � ∞ ) . if supp b ∩ supp c = ∅ , then

Let h be a nice function. Let x ( t ) , p ( t ) solve the Hamilton equa- tions with the Hamiltonian h and the initial conditions x (0) , p (0) . � � Then r t ( x (0) , p (0)) = x ( t ) , p ( t ) defines a symplectic transforma- tion. Formally, � Op( h ) = Op( b ◦ r t ) + O ( � 2 ) . − i t i t � Op( h ) Op( b )e e Under various assumptions this asymptotics can be made rigorous, and then it is called the Egorov Theorem.

If h is a quadratic polynomial, the transformation r t is linear and i t � Op( h ) there is no error term in the Egorov Theorem. The operators e generate a group, which is the double covering of the symplectic group called the metaplectic group.

If b, c ∈ L 2 ( R 2 d ) , then (rigorously) � TrOp( b ) ∗ Op( c ) = (2 π � ) − d b ( x, p ) c ( x, p )d x d p. Setting b = 1 we obtain (heuristically) � TrOp( c ) = (2 π � ) − d c ( x, p )d x d p.

Formally, Op( b ) n = Op( b n ) + O ( � 2 ) . Hence for polynomial func- tions f ◦ b + O ( � 2 ) � � � � f Op( b ) = Op . One can expect this to be true for a larger class of nice functions. Consequently, f ◦ b + O ( � 2 ) � � � � Tr f Op( b ) = TrOp � = (2 π � ) − d d x d p + O ( � − d +2 ) . � � f b ( x, p )

For a bounded from below self-adjoint operator H set N µ ( H ) := Tr1 l ] −∞ ,µ ] ( H ) , which is the number of eigenvalues ≤ µ of H counting multiplicity. Then setting f = 1 l ] −∞ ,µ ] , we obtain � = (2 π � ) − d d x d p + O ( � − d +2 ) . � � N µ Op( h ) h ( x,p ) ≤ µ In practice the error term O ( � − d +2 ) may be too optimistic and one gets something worse (but hopefully at least o ( � − d ) ).

For example, if V − µ > 0 outside a compact set, then N µ ( − � 2 ∆ + V ( x )) � d ≃ (2 π � ) − d c d − d x + o ( � − d ) , | V ( x ) − µ | 2 V ( x ) ≤ µ which is often called the Weyl asymptotics.

Aspects of quantization. Fundamental formalism – used to define a quantum theory from a classical theory; – underlying the emergence of classical physics from quantum physics. Technical parametrization – of operators used to prove theorems about PDE’s; – of observables in quantum optics and signal processing.

Elements of quantization should belong to standard curriculum! Example: standard courses at FACULTY OF PHYSICS, UNIVERSITY OF WARSAW. Quantum Mechanics 1. (nonrelativistic theory); Quantum Mechanics 1 1 2 . (quantization, quantum information); Quantum Mechanics 2A (relativistic theory); Quantum Mechanics 2B (many body theory); ....