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Where are those quanta? and How many there are? Università di Pavia 20.04.2015 Juan León QUINFOG Instituto de Física Fundamental (CSIC) 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 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| 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 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) | > strictly localized in a region | >≠ c1….N |n1……nN> Localization and Fock representation at loggerheads Unitary inequivalence by example a, b operators, c c-number In the new ‘b’ representation states and operators behave as in the old one ‘a’ 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 Finite sum of real numbers NRQM (Stone Von Neumann) (QM) Countable infinite number of degrees of freedom Examples of unitary inequivalence: Bloch Nordsieck (1937) Haag’s Theorem (1955) Unruh effect Knight Light (1962) (infrared catastrophe, electron cannot emit a finite nº of photons) (interaction pictue exists only for free fields) (Number operator not unique) (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. Particles Vacuum global d.o.f. elementary excitations of global oscillators ground state (maximum rest) All oscillators are present in the vacuum Local excitations are not particles, Global are Vacuum entanglement: what you spot at (standard Fock Space) depends on Hegerfeldt Theorem Take as (unbound spreading) analytic in 0 Im t 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 For t infinitesimal (x,t) ≠ 0 even A simplified version ofIm Ree-Schlieder th. 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 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 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 Cauchy surface t=0 r Local quanta given by Modes out of r initially localized in r Operators Two types of quanta global local eigenstates both sign frequencies cannot vanish outside finite intervals localized within finite intervals 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 ‘ t 0 L R x ´ 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 ‘ The sin 𝜋𝑘𝑥 𝑟 are complete and orthogonal in [0,r] N.B. The 𝑐 k above are not the derivatives of the 𝑐𝑘 t 0 L R x ´ 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 ‘ t 0 L R x ´ 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 UPPER CASE and build up lower case at t=0 lower case Local modes are superpositions of positive and negative frequencies t x Field expansions Bogoliubov transformations Canonical conmutation relations Local quantization Local vacuum Unitary Inequivalence Positivity of energy Energy of the local quanta: Exciting the vacuum with local quanta How much localization to expect 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 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!