Download Localization and the Semiclassical Limit in Quantum Field Theories

Localization and the Semiclassical Limit in
Quantum Field Theories
J C A Barata
Departamento de Fı́sica Matemática
Instituto de Fı́sica da Universidade de São Paulo
(joint work with Nelson Yokomizo, René S. Freire and Thiago C. Raszeja – IFUSP)
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The notion of particle states in QFT
QFT as a fundamental theory.
Compatibility between the classical notion of fields and the corpuscular
nature of matter
The contributions of Haag and Swieca (1965) and of Buchholz and
Wichmann (1986): limitations on the local degrees of freedom of
relativistic quantum fields.
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The Wigner notion of particle states (1939): irreducible representations of
the Poincaré group.
√
Particles as eigenstates of the mass operator H 2 − P 2 .
The energy momentum spectrum
E
m
p
Figura 1: The energy-momentum spectrum and E =
3
p
p2 + m2 .
This picture, however, is not valid for the electron (Buchholz - 1986).
Infraparticles, etc.
Buchholz, D. (1986). ”Gauss’ law and the infraparticle problem”. Physics
Letters B 174: 331
The more fundamental notion follows the intuitive idea that particle states
represent localized excitations that propagate in a stable well-defined way.
4
More problems:
the notion of particle for QFTs formulated in a curved space-time.
5
The existence of particles, though, in part of our macroscopic classical
experience.
Particle states should be identifiable if their classical limit (formally
characterized by ~ → 0) lead to classical particle states.
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Hepp and the Classical Limit of Quantum Systems
In 1974 Klaus Hepp performed a rigorous and detailed (non-perturbative!)
analysis of the semiclassical limit of quantum systems:
• Non-relativistic quantum systems with finite degrees of freedom
• Non-relativistic many-body systems
• Relativistic quantum field theory models. More specifically, models for
scalar fields in 1+1 Minkowski space-time (P (φ)2 ).
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Hepp made extensive use of the notion of coherent state (Schrödinger,
1926).
If for the one-dimensional harmonic oscillator with mass m with a
potential U (x) = k2 x2 we consider the initial state
!
2
mω 1/4
mω0 x − x0
0
exp −
ψ(x, 0) =
π~
~
2
ω0 =
q
k
,
m
we will have
mω 1/2
mω
2
2
0
0
ψ(x, t) =
x − x0 cos(ω0 t)
exp −
.
π~
~
For each t, it is a Gaussian centered at x0 cos(ω0 t), reproducing the
classical motion of the harmonic oscillator.
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In general, coherent states for the harmonic oscillator are given by
|αi := exp αa∗ − α∗ a |0i
with α ∈ C and
Writing α =
√1
2
1
−1
a := √ q + ~ p .
2
(ξ + iπ), we also have
|αi := exp i πq − ~−1 ξp |0i .
Coherent states satisfy
a|αi = α|αi
and are states of minimal “uncertainty”: ∆p∆q = ~/2.
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Hepp’s theorems: finite number of degrees of freedom
Consider the classical non-relativistic unidimensional motion of a particle of
p2
p
mass m with a Hamiltonian H = 2m + V (q), such that ṗ = − ∂V
,
q̇
=
.
∂q
m
Let (ξ(t), π(t)) be the solution for the initial conditions
q(0) = ξ, p(0) = π.
Consider the corresponding Hamilton operator
~2 ∂ 2
+ V (q) ,
H = −
2
2m ∂q
∂
is the momentum
where q is the position operator and p = −i~ ∂q
operator acting on some as self-adjoint operators on some suitable domain
− it
2
in L (R, dx). Let U (t) = e ~ H be the corresponding unitary propagator.
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Then, for the Weyl operators and for |t| < T , one has
lim hα| U (t)∗ ei(aq+bp) U (t) |αi = ei[aξ(t)+bπ(t)] ,
~→0
√
where |αi, with α = (ξ + iπ)/( 2~) are coherent states (parametrized by
the given classical initial conditions!).
Hence, the classical trajectory (ξ(t), π(t)) can be recovered for |t| < T .
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For ~ “small” the quantum system is ruled by the Hamilton operator
“linearized” about the classical trajectory:
′′
1 2 V ξ(α, t) 2
p +
q .
H(t) :=
2m
2
Moreover,
lim U (t)|αi − |φ(t)i .
~→0
where
|φ(t)i := exp i π(t)q − ~−1 ξ(t)p W (t)|0i
and where W (t) is the propagator associated to H(t):
Z t
W (t) := T exp −i~
H(t′ ) dt′ .
0
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In the position representation, one has
1/4
2
ω(t)
π(α, t)
ω(t)
x − ξ(α, t) + i
exp −
φ(x, t) =
x .
π~
2~
~
Generalization of this to distinguishable particles is trivial.
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Hepp’s theorems: non-relativistic many-body systems
Let us consider a bosonic many-body system of non-relativistic
indistinguishable particles described in the Fock space
Fs (H) =
∞
M
L2 (Rn , dn x)S ,
n=0
where L2 (Rn , dn x)S is the Hilbert space of symmetric square integrable
functions in Rn , with a Hamiltonian
Z
2 Z
~
1
∗
2
H = −
a (x)∇ a(x) dx+
a∗ (x)a∗ (y)V (x−y)a(x)a(y) dx dy ,
2m
2
with
∗
∗
a(x), a(y) = a (x), a (y) = 0 ,
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∗
a(x), a (y) = δ(x − y) ,
Proceeding in an analogous fashion, and adopting the coherent states,
Z
|αi := exp
[α(x)a∗ (x) − α(x)∗ a(x)] dx |0i ,
with α(x) ∈ C, we get in the ~ → 0 limit a classical field described by the
integro-differential equation
Z
∂α
1 2
i (t, x) = − ∇ α(t, x) − V (x − y)|α(t, y)|2 α(t, x) dy ,
∂t
2µ
where µ is a constant depending on m, the “mass” of the original bosonic
particles.
For V (x − y) = gδ(x − y), for instance, we get
1 2
∂α
i (t, x) = − ∇ α(t, x) − g|α(t, x)|2 α(t, x) ,
∂t
2µ
the well-known Gross–Pitaevskii equation.
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Particle States in the semiclassical limit
Hepp noticed that, depending on the coherent states chosen and of the
form in which observables and parameters are rescaled when ~ → 0, other
limit states can be reached: states describing not classical fields, but
classical N -particle systems (N being chosen freely).
However, there is a difficulty when we deal with identical particles: the
lack of observables that describe individual kinematical properties
(position, momentum, etc.)
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We choose in the N particles subspace of the Fock space symmetrized
coherent states:
1 X
|απ(1) i ⊗ · · · ⊗ |απ(N ) i
|α1 , . . . , αN iS =
N π
with |αk i being one-particle coherent states.
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For the observables we adopt localized observables
(N )
AO
:=
N
X
i=1
1 ⊗ · · · 1 ⊗ AO ⊗ 1 ⊗ · · · ⊗ 1
with
AO :=
for which we have
A χO + χO A /2 ,
hAiψ − hAO iψ ≤ kAk (1 − χO )ψ so that if ψ is strongly concentrated in the region O the difference
between the two expectation values is small.
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Consider the Weyl operators acting in the one-particle sub-space
W(a, b) := exp i(aq + bp)
Taking Oj (t) as a ball of fixed radius centered in ξj (α, t) (the position of
the j-th particle at time t) we have
(N )
lim hα1 , . . . , αN |S U (t)∗ W(a, b)Oj (t) U (t) |α1 , . . . , αN iS = ei[aξj (t)+bπj (t)] ,
~→0
with the classical dynamics described by
Hc =
X πj2
j
ξ˙j = πj ,
1X
V (ξj − ξk ) ,
+
2
2 j6=k
π˙j = −
X
k6=j
V ′ (ξj − ξk ) .
Hence, we can isolate the trajectory of the j-th particle and, therefore,
distinguish them in the semiclassical limit.
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There are, therefore, two kinds of semiclassical limits in non-relativistic
many-body systems: one describing classical fields and other describing
systems of N classical non-relativistic particles.
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Hepp’s theorems: relativistic QFT models
In the case of P (φ)2 models (in 1 + 1 Minkowski space-time) Hepp
obtained results analogous to those of non-relativistic many-body systems.
In P (φ)2 models we have:
H = H0 +
Z
N
X
dx
n=1
an : Φ(x)n :
!
for even N and aN > 0. Those models have been constructed by Glimm e
Jaffe in a series of works in the 70ies.
The classical field equation associated to the Hamiltonian above is
N
X
nan ϕ(x, t)n−1 = 0 .
✷ + m2 ϕ(x, t) +
n=1
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Similarly to non-relativistic many-body systems, with an adequate
reparametrization of the coupling constants and with a convenient choice
of coherent states, the classical dynamics of the quantized fields above
converges when ~ → 0 to the dynamics of the classical fields ϕ(x, t)
described above:
N
X
✷ + m2 ϕ(x, t) +
nan ϕ(x, t)n−1 = 0 .
n=1
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The question now is whether classical relativistic particles and their
dynamical evolution can be also recovered from the quantized bosonic
fields in the limit ~ → 0.
An important problem here is to identify appropriate localized states and
position operators in the context of relativistic quantum mechanics.
This question was analyzed and answered in a classical work by Newton e
Wigner in 1949.
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Newton and Wigner position operator
States strictly localized in a single point (like Dirac distributions) are not
possible in RQM due to the restriction to positive energy solution of the
Klein-Gordon equation.
States in RQM can only be localized in some approximate sense.
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Newton and Wigner based their analysis on the hypothesis that strictly
localized states must (by definition!) satisfy four postulates:
• The set of states localized about a point must form a linear space.
• Invariance by rotations and space-time reflections.
• Orthogonality by translations.
• Certain technical regularity conditions (reanalyzed by Wightman in
1962.)
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Two relevant one-particle Hilbert spaces (in momentum representation):
H1 = L2 (R, dp/ω), with the relativistically invariant scalar product
Z
dp
hφ|ψiH1 :=
φ(p)ψ(p) ,
R ω(p)
p
with ω(p) = p2 + m2 , and H2 = L2 (R, dp), with the usual scalar
product
Z
hφ|ψiH2 :=
R
The map M√ω : H2 → H1 defined by
(M√
dp φ(p)ψ(p)
√
ωφ(p)
ω φ)(p) :=
is unitary.
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The momentum operator in H1 and H2 is multiplication by p.
∂
The usual position operator in H2 is i~ ∂p
and its corresponding version in
H2 is
∂
√
M√−1ω .
q := M ω i~
∂p
We have
√ ∂
(qφ)(p) = i~ ω
∂p
φ(p)
√
ω
= i~
∂
p
− 2
∂p 2ω
φ(p) .
√
The eigenstate of q is w which, corresponds in coordinate space to
1/4
2m~
K1/4 (−m|x|/~) .
|x|
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Making use of this Newton-Wigner position operator, of the localized
states above and of adequate coherent states (Kaiser, 1978) we were able
to reproduce the previous results and obtain N -particle classical systems
for the quantized Klein-Gordon field.
Therefore, classical field limits and classical particle limits coexist in QFT!
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Addicional results:
Localization operators in 1 + 1 and 2 + 1 de Sitter spaces (with N.
Yokomizo and T. Raszeja, resp.).
Coherent states for massive scalar free field theories (with R. Freire).
Recovery of the (free) classical particle dynamics in de Sitter space.
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This work is associated to another larger project in collaboration with
Christian Jäkel (Cardiff University, Wales) and Jens Mund (UFJF, Brasil)
• Construction of interacting P (φ)2 models in 1 + 1 dimensional de Sitter
space.
• Construction of non-trivial nets of von Neumann Algebras describing
covariant Quantum Fields Theories in the sense of Algebraic Quantum
Field Theory, also in the 1 + 1 dimensional de Sitter space.
• Construction of non-trivial nets of von Neumann Algebras describing
covariant Quantum Fields Theories in the sense of Algebraic Quantum
Field Theory, in the 1 + 1 dimensional Minkowski space, by taking the
limit r → ∞ in the previous construction.
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Projects
Generalization for Quantum Field Theories in a general curved space-time.
Generalization for interacting quantum fields in 1 + 1 de Sitter space.
General criterion (BW) for the semiclassical limit of QFTs.
To settle the notion of particle states in QFT formulated in curved
spacetimes in terms of its semiclassical limits.
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References
• K. Hepp, “The classical limit of quantum mechanical correlation
functions”, Commun. Math. Phys. 35 (1974) 265.
• R. Haag and J. A. Swieca, “When does a quantum field theory describe
particles?”, Commun. Math. Phys. 1 (1965) 308.
• D. Buchholz and E. Wichmann, “Causal independence and the energy
level density of states in local quantum field theory”, Commun. Math.
Phys. 106 (1986) 321.
• T. D. Newton and E. P. Wigner, “Localized states for elementary
systems”, Rev. Mod. Phys. 21 (1949) 400.
• N. Yokomizo and J. C. A. B., “Multiple classical limits in relativistic and
nonrelativistic quantum mechanics”, J. Math. Phys. 50, 123512 (2009).
N. Yokomizo and J. C. A. B., “Localizability in de Sitter space”, J. Phys.
A: Math. Theor. 45 (2012) 365401.
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