Localization and the Semiclassical Limit in Quantum Field Theories

J C A Barata

Departamento de F´ısicaMatem´atica

Instituto de F´ısicada Universidade de S˜aoPaulo

(joint work with Nelson Yokomizo, Ren´eS. Freire and Thiago C. Raszeja – IFUSP)

1 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.

2 The Wigner notion of particle states (1939): irreducible representations of the Poincar´egroup. Particles as eigenstates of the mass operator √H2 P 2. − The energy momentum spectrum

Figura 1: The energy-momentum spectrum and E = p2 + m2. 3 p 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. →

6 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).

7 Hepp made extensive use of the notion of coherent state (Schr¨odinger, 1926). If for the one-dimensional harmonic oscillator with mass m with a k 2 potential U(x) = 2 x we consider the initial state

2 1/4 mω0 mω0 x x0 ψ(x, 0) = ~ exp ~ − π − 2 ! k ω0 = m , we will have q 2 mω 1/2 mω 2 ψ(x, t) = 0 exp 0 x x cos(ω t) . π~ − ~ − 0 0 For each t, it is a Gaussian centered at x0 cos(ω0t), reproducing the classical motion of the harmonic oscillator.

8 In general, coherent states for the harmonic oscillator are given by

α := exp αa∗ α∗a 0 | i − | i with α g and ∈ 1 1 a := q + ~− p . √2 Writing α = 1 (ξ + iπ), we also have √2

1 α := exp i πq ~− ξp 0 . | i − | i Coherent states satisfy a α = α α | i | i and are states of minimal “uncertainty”: ∆p∆q = ~/2.

9 Hepp’s theorems: ﬁnite number of degrees of freedom Consider the classical non-relativistic unidimensional motion of a particle of 2 mass m with a Hamiltonian H = p + V (q), such that p˙ = ∂V , q˙ = p . 2m − ∂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 H = + V (q) , −2m ∂q2 where q is the position operator and p = i~ ∂ is the momentum − ∂q operator acting on some as self-adjoint operators on some suitable domain it 2 ~ H in L ( , dx). Let U(t) = e− be the corresponding unitary propagator.

10 Then, for the Weyl operators and for t < T , one has | | i(aq+bp) i[aξ(t)+bπ(t)] lim α U(t)∗ e U(t) α = e , ~ 0 → h | | i where α , with α = (ξ + iπ)/(√2~) are coherent states (parametrized by | i the given classical initial conditions!). Hence, the classical trajectory (ξ(t), π(t)) can be recovered for t < T . | |

11 For ~ “small” the quantum system is ruled by the Hamilton operator “linearized” about the classical trajectory:

1 V ′′ ξ(α, t) H(t) := p2 + q2 . 2m 2 Moreover, lim U(t) α φ(t) . ~ 0 | i − | i → where 1 φ(t) := exp i π(t)q ~− ξ(t)p W (t) 0 | i − | i and where W (t) is the propagator associated to H(t):

t W (t) := T exp i~ H(t′) dt′ . − Z0

12 In the position representation, one has

1/4 ω(t) ω(t) 2 π(α, t) φ(x, t) = exp x ξ(α, t) + i x . π~ − 2~ − ~ Generalization of this to distinguishable particles is trivial.

13 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

∞ 2 n n ( ) = L ( , d x) , Fs H S n=0 M 2 n n where L ( , d x)S is the Hilbert space of symmetric square integrable n functions in , with a Hamiltonian ~2 2 1 H = a∗(x) a(x) dx+ a∗(x)a∗(y)V (x y)a(x)a(y) dx dy , −2m ∇ 2 − Z Z with

a(x), a(y) = a∗(x), a∗(y) = 0 , a(x), a∗(y) = δ(x y) , − 14 Proceeding in an analogous fashion, and adopting the coherent states,

α := exp [α(x)a∗(x) α(x)∗a(x)] dx 0 , | i − | i Z

with α(x) g , we get in the ~ 0 limit a classical field described by the ∈ → integro-differential equation ∂α 1 i (t, x) = 2α(t, x) V (x y) α(t, y) 2α(t, x) dy , ∂t −2µ∇ − − | | Z 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 − − ∂α 1 i (t, x) = 2α(t, x) g α(t, x) 2α(t, x) , ∂t −2µ∇ − | | the well-known Gross–Pitaevskii equation. 15 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.)

16 We choose in the N particles subspace of the Fock space symmetrized coherent states: 1 α , . . . , α = α α | 1 N iS | π(1)i ⊗ · · · ⊗ | π(N)i π N X with α being one-particle coherent states. | ki

17 For the observables we adopt localized observables

N (N)

A := I I A I I O ⊗ · · · ⊗ O ⊗ ⊗ · · · ⊗ i=1 X with

AO := A χO + χO A /2 , for which we have

A A A (1 χ )ψ h iψ − h Oiψ ≤ k k − O so that if ψ is strongly concentrated in the region O the diﬀerence between the two expectation values is small.

18 Consider the Weyl operators acting in the one-particle sub-space

(a, b) := exp i(aq + bp) W

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) i[aξj (t)+bπj (t)] lim α1, . . . , αN S U(t)∗ (a, b) U(t) α1, . . . , αN S = e , ~ 0 Oj (t) → h | W | i with the classical dynamics described by π2 1 H = j + V (ξ ξ ) , c 2 2 j − k j j=k X X6

ξ˙ = π , π˙ = V ′(ξ ξ ) . j j j − j − k k=j X6 Hence, we can isolate the trajectory of the j-th particle and, therefore, distinguish them in the semiclassical limit. 19 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.

20 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:

N n H = H0 + dx an : Φ(x) : n=1 ! Z X 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 2 n 1 ✷ + m ϕ(x, t) + nan ϕ(x, t) − = 0 . n=1 X

21 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 2 n 1 ✷ + m ϕ(x, t) + nan ϕ(x, t) − = 0 . n=1 X

22 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.

23 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.

24 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.)

25 Two relevant one-particle Hilbert spaces (in momentum representation):

1 2 = L ( , dp/ω), with the relativistically invariant scalar product H dp φ ψ 1 := φ(p)ψ(p) ,

h | iH ω(p) Z 2 2 2 2 with ω(p) = p + m , and = L ( , dp), with the usual scalar H product p

φ ψ 2 := dp φ(p)ψ(p)

h | iH Z The map M : 2 1 deﬁned by √ω H → H

(M√ωφ)(p) := √ωφ(p) is unitary.

26 The momentum operator in 1 and 2 is multiplication by p. H H The usual position operator in 2 is i~ ∂ and its corresponding version in H ∂p 2 is H ∂ 1 q := M i~ M − . √ω ∂p √ω We have ∂ φ(p) ∂ p (qφ)(p) = i~√ω = i~ φ(p) . ∂p √ω ∂p − 2ω2 The eigenstate of q is √w which, corresponds in coordinate space to

2m~ 1/4 K ( m x /~) . x 1/4 − | | | |

27 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!

28 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.

29 This work is associated to another larger project in collaboration with Christian J¨akel (Cardiﬀ University, Wales) and Jens Mund (UFJF, Brasil) Construction of interacting P (φ) models in 1 + 1 dimensional de Sitter • 2 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. → ∞

30 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.

31 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. 32