1. Elementary excitations
Spanish Quantum Information Workshop
"Espainako Informazio Kuantikoa
(ICE-2)
Información Cuántica en España" 
University of the Basque Country, Bilbao, Spain
2 June 2015
From wave functions to
Elementary excitations
Juan León
QUINFOG
Instituto de Física Fundamental (CSIC)

5. propaganda done come back to Physics
Local quanta, unitary inequivalence, and vacuum entanglement
Rodriguez-Vazquez, del Rey, Westman, Leon, Annals of Physics, 351, 112 (2014)
What does it mean for half of an empty cavity to be full?
Brown, del Rey, Westman, Leon, Dragan, Phys. Rev. D 91, (2015)

6. Juan: What’s an electron?
EM LabTeaching Assistant: This
Charged particle as (is) a point creating a Coulomb field
Classical Charged Particles (Lorentz, Abraham, Rohrlich, Dirac,..)
Particles follow world lines

7. 1923 de Broglie: electrons behave as matter waves
In his PhD thesis … diffícult to swallow … (Einstein gave OK)
1925 Schrödinger talk in Zurich on the de Broglie waves
Debye question: What is these waves’ equation?

8. classical matter waves
φ 𝑥,𝑦, 𝑡
linear
superposition
Water waves for example
oscillatory surface height
How many?
Where?

9. physical picture … is it lost?
Heisenberg each particle its own 𝑞 , 𝑝 , 𝑞,𝑝 = 𝛿
Schrödinger wavefunction in 3 𝑛 configuration space
superpositions?
physical picture … is it lost?
𝜑( 𝑞 1 ,….., 𝑞 𝑛 , 𝑡)
𝑞 𝑖+1
𝑞 𝑖
probability wave in configuration space !

10. locality incompleteness
Heisenberg each particle its own 𝑞 , 𝑝 , 𝑞,𝑝 = 𝛿
Schrödinger wavefunction in 3 𝑛 configuration space
but, speaking of the wave function,
if it describes particles:
localization non-locality or
locality incompleteness

11. one particle in V1, other particle in V2, and nothing else
Born rule: one particle localized in V
⇒𝜑 𝑥 =0 ∀ 𝑥∉ V
value of 𝜑 in 𝑥 not a local property
beyond that,
one particle in V1, other particle in V2, and nothing else
𝜑(1,2) entangled? .

12. locality incompleteness
again, speaking of the wave function,
if it describes particles:
localization non-locality or
locality incompleteness

13. received view in QFT QM view Quantum Mechanics (QM)
excellent approximation to
Relativistic Quantum Field Theory (QFT)
received view in QFT
QM view

14. Two charactistics 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

15. 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|
N.B. Some cheating here (real space io config space)

16. 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 in the vacuum

17. 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

18. | > strictly localized in a region
| > ≠  c1….N |n1……nN>
localization and Fock representation at loggerheads

19. If it is not here, is it zero? (QM) no
If it is not here, there is the vacuum (QFT)
also
we cannot tell whether or not it is here … or there
Missing tool
𝑁 𝑉 , 𝑁 𝑉 ′ ≠0
If ∃ 𝑁 𝑉 local⇒ 𝑁 𝑉 >0 (Reeh-Schlieder)
𝑁 𝑉

20. Quantum springboards (back to ‘real' space + field)
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 excitations are particles, local no (standard Fock Space)
Vacuum entanglement: what you spot at depends on

21. Hegerfeldt Theorem (unbound spreading)
Take analytic in Im t ≤0

22. Antilocality of
simplified view of Reeh-Schlieder th.
𝜓 and 𝜓 =0 in V at t = 0 ill posed Cauchy Condition

23. We localized a particle 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
this is what allows field operators to be local

24. As this is not what happens to the photon,
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= only at the pinhole ,
and the photon energy is positive
Hegerfeld + antilocality + Cauchy:
the photon wave function 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

25. Local quanta given by out of r
Cauchy surface at t=0 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

26. Upper case global, lower case local
Local modes are superpositions of positive and negative frequencies
Upper case global, lower case local t x
Light cone
Classical localized modes

27. Local quantization
in,
out
in, out
Local vacuum
(L =Local)

28. 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
the states
are strictly local = 0

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

30. Quasi-local states
difference
Green: quasi-local modes

31. It is possible to define localized modes and quantize them
acting on the vacuum they produce quasi-local excitations
strict localization only possible on a ‘local vacuum’
Fock vacuum being global
correlations between localized quanta remain finite
even at spacelike separated regions

32. Thanks for your attention!
It is possible to define localized modes and quantize them
acting on the vacuum they produce quasi-local excitations
strict localization only possible on a ‘local vacuum’
Fock vacuum being global
correlations between localized quanta remain finite
even at spacelike separated regions
Thanks for your attention!