Transcript t - digital-csic Digital CSIC

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!