Juan León QUINFOG Instituto de Física Fundamental (CSIC ) Where are those quanta? and How many there are? Università di Pavia

SUMMARY If it is not here, is it zero? NO! If it is not here, there is the vacuum but…… This leaves no way for localization

Reeh- Schlieder Th.  There are no local number operators in QFT Unruh Effect  There is no unique total number operator in free QFT Haag’s Th.  There are no total number operators in interacting QFT

Two charactistic of a particle Localizability: We shold be able to say it is HERE NOW as different of THERE NOW Countability: Beginning with one (HERE,NOW), adding another (HERE,NOW), then another … We need a notion of localizability and local number operator for particles

No state with a finite number of quanta is strictly local (Knight, Licht, 1962) Strict Localisation In QM is strictly localized within a region of space if the expectation value of any local operator O(x) outside that region is zero 0, Ex. O(x) = |x > <x|

Strict Localisation In QFT is strictly localized within a region of space if the expectation value of any local operator O(x) outside that region is identical to that of the vacuum

No state with a finite number of quanta is strictly local (Knight, Licht, 1962) Strict Localisation In QFT is strictly localized within a region of space if the expectation value of any local operator O(x) outside that region is identical to that of the vacuum

|  > strictly localized in a region  |  > ≠  c 1….N |n 1 ……n N > Localization and Fock representation at loggerheads

Unitary inequivalence by example In the new ‘b’ representation states and operators behave as in the old one ‘a’ a, b operators, c c-number

is called a improper transformation when The representations a and b are classified as unitarily inequivalent A case of unitary inequivalence

Finite vs infinite degrees of freedom (  QM) Finite sum of real numbers NRQM (Stone Von Neumann) Countable infinite number of degrees of freedom Examples of unitary inequivalence: Bloch Nordsieck (1937) (infrared catastrophe, electron cannot emit a finite nº of photons) Haag’s Theorem (1955) (interaction pictue exists only for free fields) Unruh effect (Number operator not unique) Knight Light (1962) (No state composed of a finite nº of particles can be localized) Among inertial observers in relative motion; for different times,.. For different masses,….

Quantum springboards local d.o.f. global d.o.f. Particles elementary excitations of global oscillators Vacuum ground state (maximum rest) All oscillators are present in the vacuum Global are ( standard Fock Space ) Local excitations are not particles, Global are ( standard Fock Space ) Vacuum entanglement: what you spot at depends on

unbound spreading Hegerfeldt Theorem (unbound spreading) Take as analytic in 0  Im t R. Paley and N. Wiener Theorem XII

Instantaneous spreading causality problems in RQM and QFT Prigogine: KG particle, initially localized in box splits and expand at the speed of light destructive interference at strictly non local Antilocality of A simplified version of Ree-Schlieder th.

Instantaneous spreading causality problems in RQM and QFT Prigogine: KG particle, initially localized in box splits and expand at the speed of light destructive interference at strictly non local Antilocality of A simplified version of Ree-Schlieder th. For t infinitesimal Im  (x,t) ≠ 0 even for x 

Instantaneous spreading causality problems in RQM and QFT Prigogine: KG particle, initially localized in box splits and expand at the speed of light destructive interference at strictly non local Antilocality of A simplified version of Ree-Schlieder th.

We localized a Fock state just to discover that it is everywhere Wave function and its time derivative vanishing outside a finite region requires of positive and negative frequencies

But at t=0 only at the pinhole, and the photon energy is positive ( back to Prigogine ) According to Hegerfeld + antilocality the photon will spread everywhere almost instantaneously As this is not what happens to the photon, we have to abandon Fock space for describing the photon through the pinhole What happens when a photon, produced by an atom inside a cavity, escapes through a pinhole? Eventually the photon will impact on a screen at d

r Local quanta given by out of r Modes initially localized in r Operators Two types of quanta global eigenstates cannot vanish outside finite intervals local both sign frequencies localized within finite intervals Cauchy surface t=0

Photon emerging through the pinhole as a well posed Cauchy problem with initial values for and vanishing outside the hole. 1+1 Dirichlet problem where the global space is a cavity and the initial data vanish outside an interval within the cavit y Cauchy data can be written as ´ ‘ L Light cone t x 0 R

Photon emerging through the pinhole as a well posed Cauchy problem with initial values for and vanishing outside the hole. 1+1 Dirichlet problem where the global space is a cavity and the initial data vanish outside an interval within the cavit y Cauchy data can be written as ´ ‘ L Light cone t x 0 R

Photon emerging through the pinhole as a well posed Cauchy problem with initial values for and vanishing outside the hole. 1+1 Dirichlet problem where the global space is a cavity and the initial data vanish outside an interval within the cavit y Cauchy data can be written as ´ ‘ L Light cone t x 0 R

We follow a similar procedure for the case finite Cauchy data out of the interval: Global modes for all times : ( N.B. stationary modes ) How sums of and build up at t=0 UPPER CASE lower case lower case

Light cone t x Local modes are superpositions of positive and negative frequencies

Field expansions Bogoliubov transformations Canonical conmutation relations

Local quantization Local vacuum

Unitary Inequivalence

Energy of the local quanta : Positivity of energy

Exciting the vacuum with local quanta Normalized one-local quantum state If were strictly local should be zero N.B. in the local vacuum then the states Become strictly local = 0 How much localization to expect

This is not the case in global vacuum due to vacuum correlations

In QM there are conjugate operators Q, P and conjugate representations They are unitarily equivalent (Stone Von-Neumann, Pauli). In  QM there are conjugate operators  (t,x),  (t,x) and conjugate representations They may be unitarily inequivalent (infinite degrees of freedom) (Unruh, de Witt, Fulling) Energy representations …………………………………………….. global elementary exitations Localized representations ………………………………………… local elementary excitations Quantum vacuum is global vacuum entanglement

In QM there are conjugate operators Q, P and conjugate representations They are unitarily equivalent (Stone Von-Neumann, Pauli). In  QM there are conjugate operators  (t,x),  (t,x) and conjugate representations They may be unitarily inequivalent (infinite degrees of freedom) (Unruh, de Witt, Fulling) Energy representations …………………………………………….. global elementary exitations Localized representations ………………………………………… local elementary excitations Quantum vacuum is global vacuum entanglement Thanks for your attention!