Spohn H. Dynamics of Charged Particles and their Radiation Field

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16.1 The time-adiabatic theorem 223

We look for a solution which is ε-close to H (εt).Using the identities

P(t)

P(t)P(t) = 0, (1 − P(t))

P(t)(1 − P(t)) = 0, we obtain

(εt) = H(εt) + iε[

P(εt), P(εt)]. (16.11)

To prove (16.8) one thus has to estimate the difference

(t, 0) − U

(t, 0) =−ε



dsU

(t, s)[

P(εs), P(εs)]U

(s, 0). (16.12)

While H (εt) − H

(εt) is of order ε, this is not good enough, since errors might

add up over the long times ε

−1

τ .Tomake progress we note that P[

P, P]P =

0 = (1 − P)[

P, P](1 − P), whereas P[

P, P](1 − P) = 0. Thus to improve on

(16.12) one has to exploit the time averaging, which is most easily achieved by

writing [

P, P]asatime derivative. Let us assume for a moment that the commu-

tator equation

[H (t), X (t)] = [

P(t), P(t)] (16.13)

has a bounded solution X (t). Then, using again (16.11),

(ε

−1

τ,0) − U

(ε

−1

τ,0) (16.14)

=−ε



−1

dsU

(ε

−1

τ,s)



H(εs)X (εs) − X (εs)H

(εs)



(s, 0) + O (ε)

= iε



−1



(ε

−1

τ,s)X (εs)U

(s, 0) − U

(ε

−1

τ,s)X (εs)

(s, 0)



O(ε)

= iε



−1



(ε

−1

τ,s)X (εs)U

(s, 0)



− U

(ε

−1

τ,0)ε

X(εs)U

(s, 0)



O(ε),

which implies

U

(ε

−1

τ,0) − U

(ε

−1

τ,0)≤c

(1 + τ)ε. (16.15)

The adiabatic theorem (16.8) follows from

(1 − P(τ ))U

(ε

−1

τ,0)P(0)ψ=(1 − P(τ ))U

(ε

−1

τ,0)P(0)ψ+O(ε)

=(1 − P(τ ))P(τ )U

(ε

−1

τ,0)ψ+O(ε)

O(ε), (16.16)

where (16.9) has been used.

224 States of lowest energy: dynamics

It remains to see whether the commutator equation (16.13) has a solution. Be-

cause of the spectral gap we may set

X = P

P(1 − P)(H − E)

−1

+ (H − E)

−1

(1 − P)

PP (16.17)

and verify (16.13) directly. In particular, X (t)≤g

−1



P(t), with g the width

of the gap and 

X(t)≤3g

−2



H(t)

P(t).

While undoubtedly correct the estimate (16.8) does not specify the origin of the

error. As we will explain below the order ε is not due to dispersion into all of

Rather the true solution ψ(t) is slightly tilted out of the subspace P(εt)

H.Ifthis

effect is properly taken into account, the error in (16.8) can be made smaller than

any given power ε

at the expense of adjusting the projection P(t) to the slightly

tilted projection P

(t). The second missing aspect is more of a computational na-

ture. Since (1 − P

(t))H is in essence decoupled from the relevant subspace, one

would like to have an, in our case time-dependent, effective Hamiltonian govern-

ing the solution in the subspace P

(t)H,atleast approximately. We will return to

this point below.

16.2 The space-adiabatic limit

With these preparations done we return to the Pauli–Fierz model with the slowly

varying electrostatic potential V (εx) = eφ

(εx),

H =

( p − P

− eA

)

+ H

+ V (εx) = H

+ V (εx). (16.18)

The case of a slowly varying vector potential will be discussed in section 16.6. Spin

is omitted only for notational simplicity. H acts on L

, d

x) ⊗ F .Forthe wave

functions it is convenient to use the momentum representation ψ(k, k

), also for

the electron, with the shorthand k

= (k

,λ

,... ,k

,λ

), n arbitrary, ψ(k, ∅) =

ψ(k) ⊗ . H

then has the direct integral decomposition



⊕

(k). (16.19)

We assume a small photon mass and the validity of claim 15.4. Then, for every

k, |k| < p

, H

(k) has a unique ground state ψ

, H

(k)ψ

(k, k) = E (k)ψ

(k, k).

Since in the momentum representation H

(k) is a real operator, the phase of

(k) can be chosen such that the wave function is real. In particular, using

ψ

(k), ψ

(k)

= 1, this implies ψ

(k), ∇

(k)

= 0. E(k) is separated by

a ﬁnite gap from the continuum edge E

(k). Since our aim is to demonstrate the

basic principle, we deliberately ignore the fact that the ground state band exists

only up to p

and continue as if p

=∞.Atthe cost of a suitable restriction on

16.2 The space-adiabatic limit 225

the initial state, the assumption p

=∞can be avoided. We refer to the Notes at

the end of the chapter for further explanations.

The ground state band is the subspace of wave functions of the form



f (k)ψ

(k, k), and the corresponding projection is denoted by P

. P

H is invariant

under e

−iH

,[H

, P

] = 0, and

−iH



f ψ

)(k, k) =



−iE(k)t



f (k)



(k, k). (16.20)

Thus wave functions in the ground state band propagate according to a free quan-

tum evolution with the effective energy–momentum relation E(k).

If the slowly varying potential is turned on, the subspace P

H is no longer

invariant. The in-band dynamics is modiﬁed and there are transitions to excited

states. For times which are not too long their effect remains negligible and one

expects that

−iHt



f ψ

)(k, k) =



−iH

eff



f (k)



(k, k) + O(ε), 0 ≤ t ≤ ε

−1

τ, (16.21)

where, as for the time-adiabatic theorem, the time scale is determined by the con-

dition that the electron should feel the presence of the potential V . The effective

one-particle Hamiltonian, H

eff

,isdeﬁned through the Peierls substitution

eff

= E( p) + V (εx). (16.22)

Coupling to the Maxwell ﬁeld renormalizes the kinetic energy of the quantum

particle. In particular, for small velocities we have

eff

+ V (εx). (16.23)

The mass is renormalized, but the coupling to the electrostatic potential is still

given by the bare charge e.

Let us now argue with some care that the Peierls substitution gives the correct

time evolution in the ground state band. The Hamiltonian is the one speciﬁed in

(16.18) and the relevant subspace is the ground state band P

H.Inparticular, ini-

tially P

ψ(0) = ψ(0).Byconstruction, [H

, P

] = 0 and one has to understand

the transitions between P

H and (1 − P

)H = Q

H induced by V (εx).For this

purpose we decompose into a diagonal and an off-diagonal piece as

V = V

+ V

= P

+ Q

, V

= P

+ Q

. (16.24)

It should be recalled that the time evolution must be controlled over the time span

−1

τ , τ = O(1). Thus only terms of order ε

can be ignored safely.

226 States of lowest energy: dynamics

We consider ﬁrst P

, which in the ground state band acts as

ψ)(k, k) =





V (k



)



f (k − εk



)ψ

(k), ψ

(k − εk



)

(k, k)

(16.25)

with



f (k) =ψ

(k), ψ(k)

. The Peierls substitution amounts to V (ε P

since

ψ(k, k) = i∇



f (k)ψ

(k, k) + i



f (k)ψ

(k), ∇

(k)

(k, k) (16.26)

with the second term vanishing by the argument given above. The difference is

estimated as



V (εx)P

− V (ε P

)



ψ(k, k)





V (k



)



f (k − εk



)



ψ

(k), ψ

(k − εk



)

− 1



(k, k). (16.27)

In the Taylor expansion, the ﬁrst order vanishes, since ψ

(k), ∇

(k)

= 0as

before, and the error is

O(ε

). Thus we are left with showing that V

acts as a

small perturbation only.

Since H

(k) =

(k − P

− eA

)

+ H

, one has ∇

(k) =

(k − P

−

) and, with P(k) denoting the projection onto ψ

(k),then P



⊕

kP(k),

Q(k) = 1 − P(k), and Q

= 1 − P

.Ifclear from the context, the variable “k”

will be dropped. With these conventions

V (εx)P

ψ(k) =





V (k



)P(k − εk



)ψ(k − εk



)





V (k



)P(k)ψ(k − εk



)

− ε





V (k



·∇

P(k)ψ(k − εk



) + O(ε

). (16.28)

By ﬁrst-order perturbation theory

∇

P(k) =−Q(k)(H

(k) − E(k))

−1

∇

(k)P(k) + h.c., (16.29)

h.c. denoting the Hermitian conjugate. Therefore

V (εx)P

=−iεQ

∇ P

· F(εx) + O(ε

) (16.30)

with the shorthand ∇ P



⊕

k∇

P(k) and the force F(x) =−∇V (x).

The approximate time evolution is generated by

= H

+ V

, U

(t) = e

−iH

, (16.31)

16.2 The space-adiabatic limit 227

and our goal is to compare it with the full time evolution e

−iHt

= U (t) overtimes

of order ε

−1

, i.e. to estimate the difference

U (ε

−1

t) − U

(ε

−1

t) =−i



t/ε

dsU(ε

−1

t − s)V

(s) (16.32)

with t =

O(1).

At this point we have arrived at a structure very similar to the time-adiabatic dif-

ference (16.14). V

= O(ε) and time averaging must be used. As before, the trick

is to write V

as a time derivate, i.e. as a commutator with H

,uptounavoidable

errors of order ε

.Weset

B(k) =−Q(k)(H

(k) − E(k))

−2

∇

(k)P(k). (16.33)

Then

Q(k)∇

P(k) =−[H

(k), B(k)]. (16.34)

With the shorthand B =



⊕

kB(k) one has

∇ P

· F = [H

, B] · F = [H

, B · F] − B · [H

, F] = [H

, B · F] + O(ε),

(16.35)

since [H

, F] =

( p − P

− eA

) · [ p, F] + h.c. and [p, F] =−iε∇

F(εx).It

remains to substitute H

for H

. One has [V

, B] = Q

[V, B]P

. Since B =



⊕

kB(k) and V = V (iε∇

), the commutator is of order ε, hence

∇ P

· F = [H

, B · F] + O(ε). (16.36)

On inserting in (16.32), we get

U (ε

−1

t) − U

(ε

−1

t) =−ε



t/ε

dsU(ε

−1

t − s)[H

, B · F + F · B

∗

(s)

O(ε)

=−ε



t/ε

dsU(ε

−1

t − s)U

(s)U

(−s)[H

, B · F + F · B

∗

(s) + O(ε)

= iε



t/ε

dsU(ε

−1

t − s)U

(s)

(B · F + F · B

∗

)(s) + O(ε)

= iε(B · F + F · B

∗

(ε

−1

t) − iεU (ε

−1

t)(B · F + F · B

∗

)

− iε



t/ε



U (ε

−1

t − s)U

(s)



(B · F + F · B

∗

)(s) + O(ε)

O(ε), (16.37)

228 States of lowest energy: dynamics

since

U (−s)U

(s) = iU (−s)V

(s) = O(ε) by (16.30). As in the time-

adiabatic setting the leakage out of the ground state subspace P

H is O(ε) for

times of order ε

−1

.Inaddition we have identiﬁed the effective Hamiltonian (16.22)

which approximately governs the time evolution inside P

16.3 Matrix-valued symbols

If in (16.18) a slowly varying vector potential is added through minimal coupling,

then even on a formal level the argument of the previous section breaks down.

The reason is that the ground state subspace P

H is no longer even approximately

invariant under the time evolution. There is another subspace to take its role, but it

has to be computed rather than guessed. We immediately consider the general case

(16.1) and switch to the macroscopic space scale through the substitution x for εx.

Then the Hamiltonian under study is

H =



− iε∇

− P

− eA

(x)



−

σ · (B

+ εB

(x))

+ eφ

(x) + H

. (16.38)

As before, −i∇

refers to the total momentum, A

= A

(0), B

= B

(0).

The ﬁrst step is to mold (16.38) into the canonical space-adiabatic form. For this

purpose we have to distinguish between the classical phase space variable (q, p)

and the corresponding operators, which exclusively for the purpose of sections

16.3–16.5 are denoted by



q = x,



p =−iε∇

.Tothe Hamiltonian (16.38) in the

obvious way we associate the operator-valued function (= symbol)

H(q, p) = H

(q, p) + ε H

(q, p),

(q, p) =



p − P

− eA

(q)



−

σ · B

+ eφ

(q) + H

(q, p) =−

σ · B

(q). (16.39)

For ﬁxed (q, p), H(q, p) acts as an operator on

⊗ F , C

standing for the spin

degrees of freedom. H

is called the leading symbol and H

the subleading symbol

for H because of the extra prefactor of ε in the ﬁrst line of (16.39). To a symbol

one associates an operator through the Weyl quantization, which can be thought of

as a speciﬁc prescription for ordering x and −iε∇

.Tobegeneral, let A(q, p) be

an operator-valued function with Fourier transform



A(η, ξ ),

A(q, p) = (2π)

−3



ηd



A(η, ξ )e

i(η·q+ξ · p)

. (16.40)

16.3 Matrix-valued symbols 229

The Weyl quantization of A is then simply

(A) = (2π)

−3



ηd



A(η, ξ )e

i(η·



q+ξ ·



. (16.41)

A(q, p) is an operator-valued function and

(A) is an operator on the large

Hilbert space

H = L

) ⊗ C

⊗ F .Wewill also use the notation

(A) = A(



p) =



A (16.42)

as a shorthand. Using the inverse Fourier transform in (16.40),

(A) can be writ-

ten in the form of an integral operator as

(A)ψ(x) = (2π)

−3



ξd

yA(

(x + y), εξ )e

iξ·(x−y)

ψ(y). (16.43)

Here A acts on ψ(x) which is a

⊗ F -valued wave function, ψ ∈ L

, C

⊗

F ) = L

) ⊗ (C

⊗ F ) = H, and W

(A) is an operator acting on H. Note

that f (



q) = f (x), f (



p) = f (−iε∇

) as operators. Also W

(A) being Hermitian

is equivalent to A(q, p) = A(q, p)

∗

for all (q, p).For the Weyl quantization of

H(q, p) from (16.39) one obtains simply



p) = H, (16.44)

as it should be. Thus the adiabatic evolution problem associated with (16.38) can

be written as

iε

∂

∂t

ψ(x, t) = H (x, −iε∇

)ψ(x, t) (16.45)

with the Weyl rule for the ordering of operators. Consistent with the macroscopic

space scale we switched also to macroscopic times through the substitution of t

for εt. Equation (16.45) looks like a standard Schr

odinger equation, apart from

the fact that ψ(x, t) takes values in

⊗ F and H (q, p) acts as an operator on

⊗ F .

(q, p) has a subspace of lowest energy with the corresponding projection

denoted by P(q, p). Deliberately ignoring p

< ∞, from section 15.3 we know

already that tr[P(q, p)] = 2 and

(q, p)P(q, p) = E(q, p)P(q, p) (16.46)

with the eigenvalue

E(q, p) = E( p − eA

(q)) + eφ

(q). (16.47)

One would expect that the Peierls substitution E(



p) somehow plays the role of

the effective one-particle Hamiltonian. Note that this would leave spin precession

230 States of lowest energy: dynamics

still hidden and, in fact, it will appear as the ε-order correction to the Peierls sub-

stitution E(



p).

At this stage, as for the time-adiabatic theorem, it is convenient to abstract

from the speciﬁc origin of the space-adiabatic evolution (16.45). Thereby the

general structure of space-adiabatic problems becomes visible with the bonus

of wide applicability. For simplicity

⊗ F is replaced by C

with n arbitrary.

In fact, a ﬁnite-dimensional internal Hilbert space is not essential and only al-

lows us to remain in familiar territory. We record that the Hamiltonian H (q, p) =

(q, p) + ε H

(q, p) is a matrix-valued function, assumed to be smooth in q, p.

There is a relevant subspace of physical interest with energy band E(q, p) of con-

stant multiplicity . This means that H

(q, p) has the eigenprojection P(q, p),

(q, p), P(q, p)] = 0, with tr[P(q, p)] = ,1≤ <n, such that

(q, p)P(q, p) = E(q, p)P(q, p). (16.48)

Most importantly, H

is assumed to have a spectral gap in the sense that

|E(q, p) − E

(q, p)|≥g > 0 (16.49)

for all (q, p) and all other eigenvalues E

(q, p) of H

(q, p).Asbefore, the space-

adiabatic evolution is governed by

iε

∂

∂t

ψ(x, t) = H (



p)ψ(x, t) (16.50)

with ψ(x, t) an n-spinor, i.e. the Hilbert space for the Schr

odinger equation (16.50)

is L

) ⊗ C

= H. Note that, if in (16.50) H (



p) is replaced by H (t), then

(16.50) turns into its time-adiabatic cousin (16.7) where the role of the relevant

projection P(q, p) is taken over by P(t).

The analysis of (16.50) will be carried out in such a way as to make use only of

(16.48) and (16.49) with no further assumptions at all on the spectrum of H

(q, p)

in the subspace orthogonal to P(q, p)

.For this reason we are conﬁdent that the

ﬁnal result will apply also to the Pauli–Fierz Hamiltonian.

With the more general perspective gained, one can understand why the case

= 0 can be handled by more elementary means. In that case H

(q, p) =

( p − P

− eA

)

+ eφ

(q). Thus P(q, p) depends only on p and P(



p) =

, the projection onto the ground state subspace. This suggests that also in the

general case P(



H is the adiabatically decoupled subspace. Unfortunately



= P(



p),ingeneral, although P(q, p)

= P(q, p).Onthe other hand,

as will be shown, P(



p)(1 − P(



p)) =

O(ε). Since P(



p) is Hermitian, its

spectrum is of order ε concentrated near 0 and 1. Thus, at the expense of an error

of order ε,wecan associate to P(



p) atrue projection operator





p), and





H is the adiabatically protected subspace in lowest-order approximation.

16.3 Matrix-valued symbols 231

From the example of P(



p) just discussed, it is clear that for a study of the

Schr

odinger equation (16.50) in the limit of small ε one has to understand the re-

lationship between the multiplication of symbols and the multiplication of their

Weyl quantization, which is taken up next. Let A, B be two matrix-valued func-

tions. One deﬁnes their Moyal product A#B implicitly through the condition

(A)W

(B) = W

(A#B). (16.51)

The Moyal product is best grasped in the case where the symbols are given as

formal power series,

A(q, p) =



j≥0

(q, p), B(q, p) =



j≥0

(q, p), (16.52)

where the expansion coefﬁcients A

, B

do not depend on ε. The equality is un-

derstood as |A −



n−1

j≥0

|≤c

with constants c

possibly growing so fast

in n that the partial sums in (16.52) do not converge. Then A#B also has a formal

power series, which is written as

A#B =



j≥0

(A#B)

. (16.53)

Equating power by power in (16.51) one ﬁnds

(A#B)

(q, p) =



|α|+|β|+l+m= j

(2i)

−(|α|+|β|)

(−1)

|β|

|α|!|β|!

∂

(q, p)∂

∂

(q, p),

(16.54)

where it is understood that j, l, m ∈ N and α, β are multi-indices, α, β ∈ N

.To

lowest order

(A#B)

= A

,(A#B)

= A

+ A

−

, B

}. (16.55)

We introduced here the Poisson bracket {·, ·} for matrix-valued functions. It is

deﬁned by

{A, B}=∇

A ·∇

B −∇

A ·∇

B, (16.56)

the dot referring to the scalar product of the two gradients. Thus even if the formal

power series for A, B consists only of the leading term, A = A

, B = B

,asisthe

case for P(q, p), their Moyal product is a formal power series starting with

A#B = AB − ε

{A, B}+

O(ε

) (16.57)

232 States of lowest energy: dynamics

and, by deﬁnition, the lowest-order product becomes

(A)W

(B) = W

(AB − ε

{A, B}) +

O(ε

). (16.58)

Note that in (16.56) the order of matrices must be respected. In general, it is not

true that {A, A}=0, or { A, B}=−{B, A},asone is used to from the standard

calculus of Poisson brackets.

In the sequel, very roughly the idea is to use (16.51) as a link between functions

of operators, like the time-evolved position operator



q(t) = e

iHt/ε



−iHt/ε

, and

matrix-valued symbols. In particular, one can regard the matrix-valued function

P(q, p) as the lowest-order symbol for the true Hilbert space projection onto the

adiabatically decoupled relevant subspace.

16.4 Adiabatic decoupling, effective Hamiltonians

As noticed already, in general P(



p) is not a projection, due to errors of order ε.

This suggests to successively correct P(q, p) with the goal in Weyl quantization

to get a projection up to precision ε

, n arbitrary, a situation denoted by the symbol

O(ε

∞

).Wemake the ansatz

π(q, p) =



j≥0

(q, p), π

(q, p) = P(q, p) (16.59)

and recall that in general

H(q, p) =



j≥0

(q, p), (16.60)

where in our speciﬁc application H

= 0 for j ≥ 2. The Weyl quantization for π

should be a projection and commute with H (



p) up to errors

O(ε

∞

). π has then

to satisfy the conditions

∗

= π, π#π = π, π#H = H #π. (16.61)

Through an iterative procedure it can be shown that the symbol π is in fact

uniquely determined by (16.61). By construction

(π)

= W

(π) + O(ε

∞

) and

there is a projection operator  on

H naturally associated to W

(π).Ifweassume

the initial wave function ψ to lie in 

H, ψ = ψ, then for the true solution

ψ(t) = e

−iHt/ε

ψ one has

(1 − )ψ(t) =

O(ε

∞

). (16.62)

For this reason 

H is called an almost invariant subspace, associated to the

relevant projection P(q, q).Onthe adiabatic scale transitions out of 

H are