QUADRATIC HAMILTONIANS AND THEIR RENORMALIZATION JAN DEREZI NSKI - PowerPoint PPT Presentation

QUADRATIC HAMILTONIANS AND THEIR RENORMALIZATION JAN DEREZI ´ NSKI Dep. of Math. Meth in Phys. Faculty of Physics University of Warsaw

Quadratic bosonic Hamiltonians are formally operators of the form a j + 1 j + 1 a ∗ a ∗ a ∗ ˆ � � � H := h ij ˆ i ˆ g ij ˆ i ˆ g ij ˆ a i ˆ a j + c. 2 2 Special cases: � h ii corresponds to the Weyl quantization, c = 1 2 c = 0 corresponds to the normally ordered quantization. We will see that other choices of c can be useful. Note also that if the number of degrees of freedom is infinite, c can be infinite!

One can compute the infimum of ˆ H or the vacuum energy H = 1 � �� � B 2 − E := inf ˆ B 2 4Tr + c. 0 where � � � � h − g h 0 B := , B 0 := . g − h 0 − h

I would like o discuss two examples of quadratic Hamilto- nians taken from QFT. They illustrate how nontrivial their theory can be: 1. neutral scalar field with position dependent mass, 2. charged scalar field in electromagnetic potential. These models belong to Local Quantum Physics. Even though they are not translation invariant, their dynamics is causal.

Consider the free classical neutral scalar field ( − ✷ + m 2 ) φ ( x ) = 0 , x ∈ R 1 , 3 . Together with the conjugate fields π ( x ) = ∂ t φ ( x ) they have the Poisson brackets { φ ( � x ) , φ ( � y ) } = { π ( � x ) , π ( � y ) } = 0 , x ∈ R 3 . { φ ( � x ) , π ( � y ) } = δ ( � x − � y ) , �

The Hamiltonian � � 1 x ) + 1 � 2 + 1 � 2 π 2 ( � � � 2 m 2 φ 2 ( x ) H 0 := ∂φ ( � x ) d � x 2 generates the dynamics ∂ t φ ( x ) = { φ ( x ) , H 0 } , ∂ t π ( x ) = { π ( x ) , H 0 } .

In order to diagonalize the Hamiltonian, we set � ε ( � � k 2 + m 2 k ) = and we introduce the “normal modes” �� � ε ( � � d � x k ) i (2 π ) 3 e − i � k� x a ( k ) : = 2 φ (0 , � x ) + π (0 , � x ) , � � 2 ε ( � k ) �� � ε ( � d � x k ) i � (2 π ) 3 e i � a ∗ ( k ) : = k� x 2 φ (0 , � x ) − π (0 , � x ) . � � 2 ε ( � k )

The normal modes diagonalize the Poisson relations and the Hamiltonian: { a ( k ) , a ( k ′ ) } = { a ∗ ( k ) , a ∗ ( k ′ ) } = 0 , { a ( k ) , a ∗ ( k ′ ) } = − i δ ( � k ′ ) , k − � � k ) a ∗ ( k ) a ( k ) . d � kε ( � H 0 =

The fields can be expressed in terms of normal modes as d � k � � � e i kx a ( k ) + e − i kx a ∗ ( k ) φ ( x ) = , � 2 ε ( � � (2 π ) 3 k ) � d � ε ( � k k ) � � � e i kx a ( k ) − e − i kx a ∗ ( k ) π ( x ) = . (2 π ) 3 √ � i 2

The free quantum neutral scalar field, denoted ˆ φ ( x ) , sat- isfies the hatted versions of the classical equations: ( − ✷ + m 2 )ˆ ∂ t ˆ φ ( x ) = 0 , φ ( x ) = ˆ π ( x ) . and the commutation relations [ˆ x ) , ˆ φ ( � φ ( � y )] = [ˆ π ( � x ) , ˆ π ( � y )] = 0 , [ˆ φ ( � x ) , ˆ π ( � y )] = i δ ( � x − � y ) .

The free Hamiltonian is defined in the standard way: � � 1 x ) + 1 � 2 + 1 2 m 2 ˆ � H n π 2 ( � φ 2 ( x ) ˆ � � ∂ ˆ 0 := : 2ˆ φ ( � x ) :d � x, 2 where the double dots denote the normal ordering.

a ∗ ( k ) , by the hatted classical re- One introduces ˆ a ( k ) and ˆ lations. They diagonalize the Hamiltonian and the com- mutation relations: � H n a ∗ ( k )ˆ ˆ d � kε ( � 0 = k )ˆ a ( k ) , a ( k ′ )] = [ˆ a ∗ ( k ) , ˆ a ∗ ( k ′ )] = 0 , [ˆ a ( k ) , ˆ a ∗ ( k ′ )] = δ ( � k ′ ) . k − � [ˆ a ( k ) , ˆ

Consider the classical field with a position dependent mass: ( − ✷ + m 2 ) φ ( x ) = − κ ( x ) φ ( x ) . We assume that κ is a Schwartz function. The classical Hamiltonian is � � 1 x ) + 1 � 2 + 1 m 2 + κ ( � � 2 π 2 ( � � � φ 2 ( � � � H := ∂φ ( � x ) x ) x ) d � x. 2 2

We can rewrite the Hamiltonian in terms of normal modes � d � kε ( � k ) a ∗ ( k ) a ( k ) H = d � k 1 d � k 2 κ ( � k 1 + � +1 k 2 ) � 2 � � 2 ε ( � 2 ε ( � (2 π ) 3 k 1 ) k 2 ) � � a ( − k 1 ) a ( − k 2 ) + 2 a ∗ ( k 1 ) a ( − k 2 ) + a ∗ ( k 1 ) a ∗ ( k 2 ) × .

The quantum field with a static position dependent mass satisfies ( − ✷ + m 2 )ˆ x )ˆ φ ( x ) = − κ ( � φ ( x ) . We assume that it coincides with the free field at time t = 0 . We also would like to find a Hamiltonian ˆ H such that x ) = e i t ˆ x )e − i t ˆ H ˆ ˆ H . φ ( t, � φ ( � ˆ H is defined up to an additive constant–we would like to fix a physically distinguished constant and obtain the vacuum energy.

Naively, the most obvious choice for ˆ H is the normally ordered quantization of the classical Hamiltonian: � 1 x ) + 1 � 2 + 1 � H n := 2( m 2 + κ ( � � π 2 ( � φ 2 ( � ˆ � � ∂ ˆ x ))ˆ : 2ˆ φ ( � x ) x ) :d � x 2

Expressed in terms of creation/annihilation operators it reads � H n = a ∗ ( k )ˆ ˆ d � kε ( � k )ˆ a ( k ) d � k 1 d � k 2 κ ( � k 1 + � +1 � k 2 ) � � 2 2 ε ( � 2 ε ( � (2 π ) 3 k 1 ) k 2 ) � � a ∗ ( k 1 )ˆ a ∗ ( k 1 )ˆ a ∗ ( k 2 ) × a ( − k 1 )ˆ ˆ a ( − k 2 ) + 2ˆ a ( − k 2 ) + ˆ .

H n does not exist. This can be seen when we However, ˆ try to compute the infimum of ˆ H n : the 2nd order con- tribution to the energy E 2 given by the loop with 2 ver- tices diverges. Fortunately, the higher order terms are finite, and as a Hamiltonian implementing the dynamics we could take H 2ren := ˆ H n − E 2 . ˆ Physically it is however preferable to make an additional finite renormalization, so that the counterterms are local.

Using e.g. the Pauli-Villars method we obtain the follow- ing renormalized expression for the 2nd order term: k ) | 2 d � � k E ren π ren ( � k 2 ) | κ ( � := 2 (2 π ) 3 � x ) 2 d � = E 2 − C κ ( � x, �� � � � � � k 2 + 4 m 2 k 2 + 4 m 2 + � k 2 1 π ren ( � k 2 ) := log k 2 − 2 , 4(4 π ) 2 � � � � k 2 + 4 m 2 − � � k 2 where C is an infinite counterterm.

We used π ren (0) = 0 as the renormalization condition. The physically acceptable renormalized Hamiltonian can be formally written as � H ren := ˆ H n − C ˆ x ) 2 d � κ ( � x H n − E 2 + E ren = ˆ 2 .

H ren is a well defined self-adjoint operator (despite that ˆ H n is ill defined, and E 2 , C are infinite). It is bounded ˆ from below and its infimum is E ren = E ren 2 � 1 1 + Tr ( − ∆ + m 2 + τ 2 ) κ ( − ∆ + m 2 + τ 2 ) κ 1 ( − ∆ + m 2 + τ 2 ) τ 2 d τ 1 × ( − ∆ + m 2 + κ + τ 2 ) κ 2 π.

Consider now a space-time dependent κ : R 1 , 3 ∋ ( t, � x ) �→ κ ( t, � x ) . We assume that κ is, say, a Schwartz function. The cor- responding time-dependent renormalized Hamiltonian ˆ H ren ( κ ( t )) generates the dynamics � t 2 � � H ren ( κ ( t ))d t ˆ U ( κ, t 2 , t 1 ) := Texp − i . t 1

We can also introduce the scattering operator t →∞ e i t ˆ 0 U ( κ, t, − t )e i t ˆ H n H n S ( κ ) := lim 0 . The scattering operator satisfies the Bogoliubov identity, which expresses the Einstein causality: if supp κ 2 is later than supp κ 1 , then S ( κ + κ 1 + κ 2 ) = S ( κ + κ 2 ) S ( κ ) − 1 S ( κ + κ 1 ) .

Consider the free charged scalar classical field ψ ( x ) ( − ✷ + m 2 ) ψ ( x ) = 0 . ψ ∗ ( x ) denotes its complex adjoint. The conjugate field is η ( x ) := ∂ t ψ ( x ) . The zero time Poisson brackets are { ψ ( � x ) , ψ ( � y ) } = { ψ ( � x ) , η ( � y ) } = { η ( � x ) , η ( � y ) } = 0 , x ) , η ∗ ( � y ) } = { ψ ∗ ( � { ψ ( � x ) , η ( � y ) } = δ ( � x − � y ) .

The Hamiltonian � � � η ∗ ( � ∂ψ ∗ ( � x ) + m 2 ψ ∗ ( � x ) + � x ) � H 0 = x ) η ( � ∂ψ ( � x ) ψ ( � x ) d � x. generates the dynamics ∂ t ψ ( x ) = { ψ ( x ) , H 0 } , ∂ t η ( x ) = { η ( x ) , H 0 } .

One introduces normal modes: � �� ε ( � p ) d � x i � e − i � p� x a ( p ) = 2 ψ (0 , � x ) + η (0 , � x ) (2 π ) 3 , � � 2 ε ( � p ) � �� ε ( � p ) i d � x � a ∗ ( p ) = 2 ψ ∗ (0 , � η ∗ (0 , � e i � p� x x ) − x ) (2 π ) 3 , � � 2 ε ( � p ) � �� ε ( � p ) i d � x � 2 ψ ∗ (0 , � η ∗ (0 , � e − i � p� x b ( p ) = x ) + x ) (2 π ) 3 , � � 2 ε ( � p ) � �� ε ( � p ) i d � x � b ∗ ( p ) = e i � p� x 2 ψ (0 , � x ) − η (0 , � x ) (2 π ) 3 . � � 2 ε ( � p )

Normal modes diagonalize the Hamiltonian and the Pois- son brackets � a ∗ ( p ) a ( p ) + b ∗ ( p ) b ( p ) � � H 0 = d � pε ( � p ) , { a ( p ) , a ∗ ( p ′ ) } = { b ( p ) , b ∗ ( p ′ ) } = − i δ ( � p ′ ) , p − �

We can express fields in terms of normal modes: d � p � � � e i px a ( p ) + e − i px b ∗ ( p ) ψ ( x ) = , � (2 π ) 3 � 2 ε ( � p ) � d � p ε ( � p ) � � � e i px a ( p ) − e − i px b ∗ ( p ) η ( x ) = . (2 π ) 3 √ � i 2