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Outline
Quantum Mechanics from Periodic Dynamics
Bosonic Case
Donatello Dolce
Mainz, Germany
Starting point
(old or semi-classical formulation)
Free bosonic waves are periodic with angular frequency ω̄ =
The energies of the related bosons are Ē = ~ω̄ = Tht .
Based on: ”Compact time and determinism for bosons”
D. Dolce, arXiv:0903.3680.
2π
Tt .
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
Conclusions
Relativistic dynamics
Relativistic differential system
Minkowski metric : c 2 ds 2 = c 2 dt 2 − dx2 → Lorentz transf.
Relativistic dispersion R relation : E 2 (p) = |p|2 c 2 + M 2 c 4
2 2
Klein-Gordon action : → Euler-Lagrange: ∂µ ∂ µ + M~2c Φ(x, t) = 0
Boundary Conditions:
′
([δΦ(x, t)∂t Φ(x, t)]tt ′ +Tt = 0)
Standard Field Theory:
Dirichlet BCs at spatial infinities: Φ(∞, t) = 0
Fixed values at initial and final time: δΦ(x, t ′ ) = δΦ(x, t ′ + Tt ) ≡ 0
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
Conclusions
Relativistic dynamics
Relativistic differential system
Minkowski metric : c 2 ds 2 = c 2 dt 2 − dx2 → Lorentz transf.
Relativistic dispersion R relation : E 2 (p) = |p|2 c 2 + M 2 c 4
2 2
Klein-Gordon action : → Euler-Lagrange: ∂µ ∂ µ + M~2c Φ(x, t) = 0
′
([δΦ(x, t)∂t Φ(x, t)]tt ′ +Tt = 0)
Boundary Conditions:
The symmetries of the KG action allow periodicity Tt = 2πRt :
Φ(x, t) = Φ(x, t + 2πRt )
t∈R
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
Conclusions
Relativistic dynamics
Relativistic differential system
Minkowski metric : c 2 ds 2 = c 2 dt 2 − dx2 → Lorentz transf.
Relativistic dispersion R relation : E 2 (p) = |p|2 c 2 + M 2 c 4
2 2
Klein-Gordon action : → Euler-Lagrange: ∂µ ∂ µ + M~2c Φ(x, t) = 0
Boundary Conditions:
′
([δΦ(x, t)∂t Φ(x, t)]tt ′ +Tt = 0)
The symmetries of the KG action allow periodicity Tt = 2πRt :
Φ(x, t ′ ) = Φ(x, t ′ + 2πRt )
t ∈ S1Rt ≡
t ∈ [t ′ , t ′ + 2πRt ]
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
(ds 2 ≡ 0)
Massless bosonic field
Eigenmodes decomposition
P
P
Φ(x, t) = n Φn (x, t) = n An an e −i (ωn t−kn ·x) , ωn =
=
n
Rt
ω
(c 2 dt 2 = dx2 )
Decompactification 3+1D → 3D.
S[Tt ] =
Conclusions
Z
Z t ′ +Tt
1
dt [∂µ Φ(x, t)∂ µ Φ(x, t)]
d 3x
2
′
t
Z
X
Tt
ω2
3
d x
∂i Φn (x)∂ i Φn (x) + 2n Φ2n (x)
2
c
n
ωn = nω̄
Induced spatial periodicity: λx = Tt c = h/|p̄|.
Quantized spectrum
En2 = ~2 ωn2 = n2 ~2 ω̄ 2 ;
(~ = 6.58 × 102 MeV s)
p2n = ~2 k2n = n2 ~2 ω̄ 2 c 2
ω̄ =
1
Rt
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
(ds 2 ≡ 0)
Massless bosonic field
Eigenmodes decomposition
P
P
Φ(x, t) = n Φn (x, t) = n An an e −i (ωn t−kn ·x) , ωn =
=
n
Rt
E
(c 2 dt 2 = dx2 )
Decompactification 3+1D → 3D.
S[Tt ] =
Conclusions
Z
Z t ′ +Tt
1
dt [∂µ Φ(x, t)∂ µ Φ(x, t)]
d 3x
2
′
t
Z
X
Tt
E2
3
d x
∂i Φn (x)∂ i Φn (x) + 2 n 2 Φ2n (x)
2
~ c
n
En = nĒ
Induced spatial periodicity: λx = Tt c = h/|p̄|.
Quantized spectrum
En2 = ~2 ωn2 = n2 ~2 ω̄ 2 ;
(~ = 6.58 × 102 MeV s)
p2n = ~2 k2n = n2 ~2 ω̄ 2 c 2
Ē = |p̄|c
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
(ds 2 ≡ 0)
Massless bosonic field
Eigenmodes decomposition
P
P
Φ(x, t) = n Φn (x, t) = n An an e −i (ωn t−kn ·x) , ωn =
=
n
Rt
E
(c 2 dt 2 = dx2 )
Decompactification 3+1D → 3D.
S[Tt ] =
Conclusions
Z
Z t ′ +Tt
1
dt [∂µ Φ(x, t)∂ µ Φ(x, t)]
d 3x
2
′
t
Z
X
Tt
E2
3
d x
∂i Φn (x)∂ i Φn (x) + 2 n 2 Φ2n (x)
2
~ c
n
En = nĒ
Induced spatial periodicity: λx = Tt c = h/|p̄|.
(~ = 6.58 × 102 MeV s)
Dynamical compactification
Rt (p̄) ≡
1
~
=
ω̄(p̄)
Ē (p̄)
,
ω̄(p̄) =
|p̄|c
~
Ē = |p̄|c
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
(ds 2 ≡ 0)
Massless bosonic field
Eigenmodes decomposition
P
P
Φ(x, t) = n Φn (x, t) = n An an e −i (ωn t−kn ·x) , ωn =
Decompactification 3+1D → 3D.
Rt (p̄)
Conclusions
n
Rt
(c 2 dt 2 = dx2 )
E
En (p̄)
En = nĒ
0
p̄
p̄
0
(~ = 6.58 × 102 MeV s)
Dynamical compactification
1
~
=
Rt (p̄) ≡
ω̄(p̄)
Ē (p̄)
,
|p̄|c
ω̄(p̄) =
~
Ē = |p̄|c
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
(ds 2 ≡ 0)
Massless bosonic field
Eigenmodes decomposition
P
P
Φ(x, t) = n Φn (x, t) = n An an e −i (ωn t−kn ·x) , ωn =
Decompactification 3+1D → 3D.
Rt (p̄)
Conclusions
~→0
n
Rt
(c 2 dt 2 = dx2 )
E
En (p̄)
p̄
0
p̄
0
(~ = 6.58 × 102 MeV s)
Dynamical compactification
Rt (p̄) ≡
1
~
=
ω̄(p̄)
Ē (p̄)
,
ω̄(p̄) =
|p̄|c
~
Relativistic Limit
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
(ds 2 ≡ 0)
Massless bosonic field
Eigenmodes decomposition
P
P
Φ(x, t) = n Φn (x, t) = n An an e −i (ωn t−kn ·x) , ωn =
Decompactification 3+1D → 3D.
Rt (p̄)
Conclusions
n
Rt
~→∞
(c 2 dt 2 = dx2 )
E
En (p̄)
p̄
0
p̄
0
(~ = 6.58 × 102 MeV s)
Dynamical compactification
Rt (p̄) ≡
1
~
=
ω̄(p̄)
Ē (p̄)
,
ω̄(p̄) =
|p̄|c
~
Ē = ~ω̄
Quantum Limit
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
Conclusions
(ds 2 6= 0)
Massive bosonic field
Induced periodicity on s ( c 2 ds 2 = c 2 dt 2 − dx2 )
zero spatial separation (zero momentum): ds 2 = dt 2 .
E
5D massless field
2
2
2
2
2
2
(dS = c dt − dx − c ds ≡ 0)
Virtual Extra Dimension s with periodicity Ts = Tt (0).
Z
2 2
X Ē 2
Tt
3
2
2
2
2 M̄ c
2
S=
d x
n 2 2 Φn − (∂i Φn ) − n
Φn
2
~ c
~2
n
M̄c 2
1−(v(p̄)/c)2
Lorentz Transf.: ω̄(p̄) = √
~
Dynamical compactification
M̄c 2 ≡
~
Rt (0)
,
ω̄(p̄) =
1
Rt (p̄)
=
En = nĒ
M̄v(p̄)
,
1−(v(p̄)/c)2
k̄ = √
,
~
√
p̄2 c 2 +M̄ 2 c 4
~
Ē = ~ω̄
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
(ds 2 6= 0)
Massive bosonic field
Electron Compton Length: Rs c =
Proper Time Period.: Ts =
5D massless field
Rt (p)
Rs = Rt (0) =
h
M̄c 2
Conclusions
~
M̄c
∼ 2 · 10−12 m
∼ 8 · 10−21 s.
TCs133 ∼ 10−10 s.
Exp: Gouanère:2008
deBroglie:1924
E
(dS 2 = c 2 dt 2 − dx2 − c 2 ds 2 ≡ 0)
En (p)
~
M̄c 2
En = nĒ
p̄
0
p̄
0
Dynamical compactification
M̄c 2 ≡
~
Rt (0)
,
ω̄(p̄) =
1
Rt (p̄)
=
√
p̄2 c 2 +M̄ 2 c 4
~
Ē = ~ω̄
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
Conclusions
Causality and time ordering
Rt (p̄) is a local and dynamical property as the energy: Ē (p̄) =
~
Rt (p̄)
Green functions and retarded potential
t
−∂µ ∂ µ G (x, t; x′ , t ′ ) = 4πδ 3 (x − x′ )δ(t − t ′ ) ,
Φ(x, t) =
Z
t2
dt
′
Z
∞
3 ′
′
′
′
ω̄pt (p̄′ )
d/c
′
d x G (x, t; x , t )j(x , t )+...
interaction
−∞
t1
ω̄pt (0)
Turning on a source in the origin
ω̄γ (p̄γ )
⇒ retarded energy variation in d
d
⇒ retarded Tt variation in d: ~ω̄out = ~ω̄in + ~ω̄γ
Tt (p′ )
Tt (0)
t
interaction
before
after
Periodic BCs respect Special Relativity (Lorentz Transf., causality. . . )
x
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
Field definition and properties
Conclusions
(1+1D)
On-shell standing waves
P
P
Φ(x, t) = n An an φn (x)un (t) = n An an e −i (ωn t−kn x)
Hilbert Space
R λx dx ∗
0 λx φn (x)φm (x) = δn,m ,
hφ|χi ≡
Z
λx
0
1
λx
P
n
φ∗n (x)φn (y ) =
dx ∗
φ (x)χ(x) .
λx
P
n′
δ(x − y + λx n′ )
φn (x)
√
= hx|φn i
λx
Schrödinger equation
(∂t2 + ωn2 (p̄))un (t) = 0
i∂t φn (x)un (t) = ωn (p̄)φn (x)un (t)
i~∂t Φn (x, t) = En (p̄)Φn (x, t)
,
(∇2 + kn2 )φn (x) = 0
,
,
−i∂x φn (x)un (t) = kn φn (x)un (t)
−i~∂x Φn (x, t) = pn Φn (x, t)
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
Field definition and properties
Conclusions
(1+1D)
On-shell standing waves
P
P
Φ(x, t) = n An an φn (x)un (t) = n An an e −i (ωn t−kn x)
Hilbert Space
R λx dx ∗
0 λx φn (x)φm (x) = δn,m ,
hφ|χi ≡
Z
λx
0
1
λx
P
n
φ∗n (x)φn (y ) =
dx ∗
φ (x)χ(x) .
λx
Hamiltonian and time evolution operator,
P
n′
δ(x − y + λx n′ )
φn (x)
√
= hx|φn i
λx
Nielsen:2006
Ĥ |φn i = ~ωn |φn i ,
p̂ |φn i = ~kn |φn i
X
′
i
|φ(t)i =
U(t, t ′ ) ≡ e − ~ Ĥ(t−t ) .
e −i ωn t an |φn i = U(t) |φ(0)i ,
n
Markovian: U(t ′′ , t ′ ) =
QN−1
m=0
U(t ′ + (m + 1)ǫ, t ′ + mǫ − ǫ).
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
Path Integral
(1+1D)
Elementary time evolution
i
′
U(t ′ , t) = e − ~ Ĥ(t−t ) ,
√
U(t ′′ , t ′ ) =
√
Complete set
R λx dx ∗
φ (x)φm (x) = δn,m ,
0 λx n
Path Integral !
Conclusions
1
λx
P
n
QN−1
m=0
U(t ′ + (m + 1)ǫ, t ′ + mǫ − ǫ).
φ∗n (x)φn (y ) =
⇓
P
n
δ(x − y + λx n)
No further assumptions than periodicity: all the ingredients to build
formally a PI are already contained in the Theory.
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
Path Integral
Conclusions
(1+1D)
Feynman Path Integral
U(∆xTot , ∆tTot ) = lim
N→∞
(Time independent hamiltonian)
Z
λx
0
N−1
Y
dxm
m=1
Elementary paths are on-shell,
! (N−1
Y
m=0
hφ|e
− ~i (Ĥ∆ǫm −p̂∆xm )
|φi .
Feynman:1942.
Elementary evolution
(Φ̂ unitary field: An an ≡ 1 ∀n)
U(∆xm , ∆ǫm ) =
X
i
e − ~ (Enm ∆ǫm −pnm ∆xm ) =
nm
i
= Φ̂(∆xm , ∆ǫm ) = λx hφ|e − ~ (Ĥ∆ǫm −p̂∆xm ) |φi
∆tTot = t ′′ − t ′ ,
∆ǫm = ǫm+1 − ǫm = ǫ,
)
∆xTot = x ′′ − x ′
∆xm = xm+1 − xm
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
Path Integral
Conclusions
(1+1D)
Periodic Path Integral
U(∆xTot , ∆tTot ) = lim
N→∞
Z
0
λx
N−1
Y
dxm
λx
m=1
Poisson summation
U(∆xm , ∆ǫm )
(
=
X
! (N−1
YX
e
− ~i (Enm ∆ǫm −pnm ∆xm )
m=0 nm
P
n
e −i αn = (2π)
e −i nm (ω̄(p̄)∆ǫm −k̄∆xm )
P
n′
δ(α + 2πn′ ))
nm
= Φ̂(∆xm , ∆tm )
= 2π
X
nm
)
δ ω̄(p̄)∆τm − k̄∆xm + 2πnm
Self Interference: this mathematical integration procedure will be
interpret in terms of sum over Periodic Paths
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
Summing over Periodic Paths
Path Integral
U(∆xTot , ∆tTot ) = 2π
X
n
δ ω̄(p̄)∆tTot − k̄∆xTot + 2πn) .
t/Tt
2.5
2.0
(1.5, 3.4)
1.5
1.0
0.5
0
1
2
3
4
5
x/λx
PI as the sum over paths with different winding numbers . . .
Conclusions
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
Conclusions
Summing over Periodic Paths
Path Integral
U(∆xTot , ∆tTot ) = 2π
X
n
δ ω̄(p̄)∆tTot − k̄∆xTot + 2πn) .
2.5
t/Tt
2.0
(1.5, 3.4)
1.5
1.0
0.5
x/λx
0
1
2
3
4
5
that can be combined to form the variation around a Classical Path . . .
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
Summing over Periodic Paths
Path Integral
U(∆xTot , ∆tTot ) = 2π
X
n
t/Tt
δ ω̄(p̄)∆tTot − k̄∆xTot + 2πn) .
5
4
(3.4, 3.4)
3
2
1
x/λx
0.0
0.5
1.0
1.5
2.0
2.5
3.0
. . . preserving the variational principle!
Conclusions
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
Conclusions
Periodic Paths Interference
5
t/Tt
Constructive
self-interference along the
relativistic particle path.
4
(3.5, 3.4)
(3.4, 3.4)
3
2
1
x/λx
0.0
Periodic Wave:
Φ(x, t) =
N
X
an e −i (ωn t−kn x)
n=0
Coherent weight:
1.0
an = e α
2 /2 αn
√
n!
1.5
(3.5, 3.4)
|Φ(x, t)|
2.0
2.5
3.0
N = 15
(3.4, 3.4)
2
0.5
n
α
an = e α /2 √
n!
2
0.5
0.5
t/Tt
x/λx
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
Conclusions
Non-relativistic (massive) free particle
Non-relativistic limit
(p̄ ≪ M̄c)
q
1 p̄2
Ept (p̄) = Ē (p̄) = M̄ 2 c 4 + p̄2 c 2 ∼ M̄c 2 +
2 M̄
Free particle
(n = 1)
Upt (∆xTot , ∆tTot ) =
Z
λx
×
∼
N−1
Y
N−1
Y
m=1
dxm
λx
!
×
−i Ē (p̄)ǫm + i p̄(xm − xm−1 )
exp
~
m=0
M̄c 2 ′′
M̄ (x ′′ − x ′ )2
′
(t − t ) + i
exp −i
~
~ 2(t ′′ − t ′ )
Wave/Particle duality and double slit experiment
E
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
Heisenberg Uncertainty Relation
Wave indetermination
Ē t
e −i ~ +2πi = e −i
E (t+∆t)
~
∆t = 2π~/Ē ,
∆E × ∆t =
= e −i
(Ē +∆E )t
~
p̄
∆E = 2π~/t
(2π~)2
(2π~)2
≥
= 2π~ = h
Ē t
Ē Tt
Heisenberg Relation,
Nielsen:2006
Tt (p̄)
∆E × ∆t ≥ h
Bohr-Sommerfeld condition: En Rt = n~ .
Exact solution of the Quantum Harmonic
Oscillator and others Schrödinger problems.
Superconductivity from Gauge Symmetry
Breaking mechanism by BCs of XDs
ω(p̄)
Conclusions
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
Conclusions
Commutation Relations
O(x) Expectation value
hχ(xf , tf )|O(x)|φ(xi , ti )i =
Z
λx
dx X ∗
χ (xf , tf )e −ikm x O(x)e ikn x φn (xi , ti ) .
λx n,m m
O(x) = ∂x F (x), Integrating by parts and supposing F (x) ≡ x
hχ(xf , tf )|1|φ(xi , ti )i =
i
hχ(xf , tf )|p̂x − x p̂|φ(xi , ti )i
~
⇓
Commutation Relation
[x, p̂] = i~
∼ Canonical formulation of Quantum Mechanics
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
Conclusions
Stroboscopic quantization and t’Hooft determinism
Stroboscopic Quantization.
|N − 1i |0i
Elze:2002-2003
|n − 2i
Fast Dynamics (Tt . 10−21 s)
|1i
|2i
Time deconstruction
(Tt = Nǫ)
Discretization of the Compact time on a littice
with N sites, Georgi:2001, Berezhiani:2002.
i
Particle on a circle: U∆t=τ = e − ~ Ĥτ
|n′ i
|ni
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
Conclusions
Stroboscopic quantization and t’Hooft determinism
Stroboscopic Quantization.
|N − 1i |0i
Elze:2002-2003
|n − 2i
Fast Dynamics (Tt . 10−21 s)
|1i
|2i
Time deconstruction
(Tt = Nǫ)
Discretization of the Compact time on a littice
with N sites, Georgi:2001, Berezhiani:2002.
i
Particle on a circle: U∆t=τ = e − ~ Ĥτ
|n′ i
|ni
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
Conclusions
Stroboscopic quantization and t’Hooft determinism
Stroboscopic Quantization.
|N − 1i |0i
Elze:2002-2003
|n − 2i
Fast Dynamics (Tt . 10−21 s)
|1i
|2i
Time deconstruction
(Tt = Nǫ)
Discretization of the Compact time on a littice
with N sites, Georgi:2001, Berezhiani:2002.
i
Particle on a circle: U∆t=τ = e − ~ Ĥτ
|n′ i
|ni
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
Conclusions
Stroboscopic quantization and t’Hooft determinism
Stroboscopic Quantization.
Fast Dynamics (Tt . 10
−21
Elze:2002-2003
s)
|N − 1i |0i
Time deconstruction
(Tt = Nǫ)
Discretization of the Compact time on a littice
with N sites, Georgi:2001, Berezhiani:2002.
|n − 2i
|1i
|2i
i
Particle on a circle: U∆t=τ = e − ~ Ĥτ
t’Hooft determinism,’tHooft:2001-2007
1
Ĥ|ni ∼ ~ω̄ n +
|ni
2
No Local-Hidden-Variables
No Bell’s Theorem or similar.
|n′ i
|ni
DETERMINISM!
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
Conclusions
Simple Interaction Picture
Perturbation theory from Periodic PI.
t
Coulomb scattering
2
hνγ (p̄γ ′ ) + M(0)c
1
1
+
Rt (p̄γ ′ ) Rt (0)
ω̄pt (p̄pt ′′ )
=
=
hνγ (p̄γ ′′ ) + M(p̄pt ′′ )c
1
1
+
Rt (p̄γ ′′ ) Rt (p̄pt ′′ )
∆λ = cTs (1 − cos θ)
FT in Curved Space-Time
Z T
Z λx
√
d 3x
dt −g L(x, t).
S∼
0
0
Interaction ∼ Variation of periodicity Tt .
ω̄γ (p̄γ ′′ )
2
interaction
region
ω̄γ (p̄γ ′ )
ω̄pt (0)
x
interaction region
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
Conclusions
Thermal QCD
Satz:2008
Exponential dilatation: dt 2 → dt
′
′
2
′
′
′
= e 2ks dt 2 , dx 2 → dx 2 = e 2ks dx2
′
′
AdS metric: 0 = dS 2 = e −2ks (dt 2 − dx 2 ) − ds
′
Temperature
2
=
′
′
dt 2 −dx 2 −dz 2
(kz)2
QCD
Thermal QCD
Hot
s
T
T ′ ∼ e −ks T
Bjorken hydrodynamical model
fields at high temperature.
Cold
T′
Magas:2003:
↔
b
QGP as a volume of massless
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
Conclusions
Thermal QCD
Satz:2008
Exponential dilatation: dt 2 → dt
′
′
2
′
′
′
′
AdS metric: 0 = dS 2 = e −2ks (dt 2 − dx 2 ) − ds
′
Energy
Thermal QCD
s → z = e ks /k
~ω
2
=
′
′
~ω ∼ e
−ks
~ω
Bjorken hydrodynamical model
exponentially.
IR
~ω ′
Magas:2003:
↔
′
dt 2 −dx 2 −dz 2
(kz)2
QCD
UV
′
= e 2ks dt 2 , dx 2 → dx 2 = e 2ks dx2
π0
π+
K+
K−
π−
the energy of the QGP decays
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
Conclusions
Thermal QCD
Satz:2008
Exponential dilatation: dt 2 → dt
′
′
2
′
′
′
′
AdS metric: 0 = dS 2 = e −2ks (dt 2 − dx 2 ) − ds
′
2
′
QCD
Thermal QCD
s → z = e ks /k
R=
1
Λ
IR
′
R =
′
′
(dt 2 − d x 2 ) ∼ e ks (dt 2 − d x2 )
′
dt 2 −dx 2 −dz 2
(kz)2
=
UV
′
= e 2ks dt 2 , dx 2 → dx 2 = e 2ks dx2
↔
e+
g
q
g
_
q
q
e
_
q
1
µ
the exponential dilatation of the periodicity is encoded in the AdS metric.
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
(βRG =
√
ds −g LA|IR =Â
Σ1
i
R d3p h µ
2
Â
Â
Π
(p
)
= 12 (2π)
µ Holo
2
S 5DYM ∼
R
d4p
(2π)2
R Σ2
∂z Aµ (p, z)|UV = 0
=
Nc
12π 2 )
h
i
2
q2
log Λq 2
ΠHolo (q) ∼ − 2kg
2
5
1
α2s,eff (q)
Âµ (p)
UV
→
1
kg52
Conclusions
∼
1
α2s
−
4π
[log Λq ],
g52 k
Âµ (p)
QCD
Λ ≫ |q| ≫ µ
z
IR
→
Aµ (p, y)|IR = Âµ (p)
R=
1
Λ
ΠHolo (p 2 )
R′ =
Âµ (p)
1
µ
Âµ (p)
ΠV (p 2 )
In the effective limit one can match with good approximation
ΠHolo (q 2 ) ∼ ΠVV (q 2 ). Thus quantum runnings of αs , WSR,
meson/barion masses and couplings, QCD spectral functions, . . . .
Pomarol:2000, ArkaniHamed:2000, Son:2003, Erlich:2006.
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
Conclusions
Conclusions
HYPOTHESIS
Every spatial point has its own clock with minkowskian time periodicity
Tt (p̄) ≡
h
Ē (p̄)
S. Relativity Lorentz Trans., Dispersion Relations, Causality ...
Quantum Energy Quant., Hilbert Sp., Schrödinger eq.,
Commutation relations, Path Integral, Uncertain relation...
Phenomena Black Body, Double Slit, Quantum HO, Supercond.,
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
Conclusions
Conclusions
HYPOTHESIS
Every spatial point has its own clock with minkowskian time periodicity
Tt (p̄) ≡
h
Ē (p̄)
S. Relativity Lorentz Trans., Dispersion Relations, Causality ...
Quantum Energy Quant., Hilbert Sp., Schrödinger eq.,
Commutation relations, Path Integral, Uncertain relation...
Phenomena Black Body, Double Slit, Quantum HO, Supercond.,
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
Conclusions
Conclusions
HYPOTHESIS
Every spatial point has its own clock with minkowskian time periodicity
Tt (p̄) ≡
h
Ē (p̄)
S. Relativity Lorentz Trans., Dispersion Relations, Causality ...
Quantum Energy Quant., Hilbert Sp., Schrödinger eq.,
Commutation relations, Path Integral, Uncertain relation...
Phenomena Black Body, Double Slit, Quantum HO, Supercond.,
NEW SCENARIO ?
Special Relativity and Quantum Mechanics
unified in a deterministic wave theory.
Field theory in compact time
Quantum Behaviors
Determinism
Toward interactions
Conclusions
Time & Periodicity
Time is defined by supposing periodic phenomena.
Operative definition of Time [SI]
A second is the duration of 9,192,631,770 periods of the
radiation corresponding to the transition between the two
hyperfine levels of the ground state of the Cs 133 atom.
Relativistic Clock
”By a clock we understand anything characterized by a
phenomenon passing periodically through identical phases so
that we must assume, by the principle of sufficient reason, that
all that happens in a given period is identical with all that
happens in an arbitrary period.”
A. Einstein (1910)
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