Decay law
in
Quantum Mechanics
and
Quantum Field Theory
Institute of nuclear physics (IFJ PAN)
Cracow, 24/10/2013
Francesco Giacosa
Outline
1. Decay law: general properties, Zeno effect, experimental evidence,
GSI anomaly
2.
Lee Hamiltonian: a QFT-like quantum mechanical approach
3. Decays in Quantum Field Theory
Francesco Giacosa
Part 1: General discussion
and exp. evidence
Francesco Giacosa
Exponential decay law
• N 0 : Number of unstable particles at the time t = 0.
N (t ) = N 0 e − Γt , τ = 1/Γ mean lifetime
Confirmend in countless cases!
• For a single unstable particle:
p(t ) = e − Γt
is the survival probability for a single unstable particle created at t=0.
(Intrinsic probabilty, see Schrödinger´s cat).
For small times:
p (t ) = 1 − Γt + ...
Francesco Giacosa
Basic definitions
Let S be an unstable state prepared at t = 0 .
Survival probabilty ammplitude at t > 0 :
a(t ) = S e −iHt S
(ℏ = 1)
Survival probability : p(t ) = a(t )
Francesco Giacosa
2
Deviations from the exp. law at short times
Taylor expansion of the amplitude:
t2
a(t ) = S e
S = 1 − it S H S −
S H 2 S + ...
2
t2
*
−iHt
a (t ) = S e
S = 1 + it S H S −
S H 2 S + ...
2
− iHt
It follows:
2
p(t) = a(t) = a (t)a(t) = 1 − t
where τZ =
*
2
(SH
1
SH S − SHS
2
2
2
S − SHS
2
)
t2
+ ... = 1 − 2 + ...
τZ
.
p(t) decreases quadratically (not linearly);
no exp. decay for short times.
τZ is the `Zeno time´.
Francesco Giacosa
Time evoluition and energy distribution (1)
The unstable state S is not an eigenstate of the Hamiltonian H.
Let dS (E) be the energy distribution of the unstable state S .
Normalization holds:
∫
+∞
−∞
dS (E)dE = 1
+∞
a(t) = ∫ dS (E)e dE
−iEt
−∞
In stable limit : d S ( E ) = δ ( E − M 0 ) → a (t ) = e − iM t → p (t ) = 1
0
Francesco Giacosa
Time evoluition and energy distribution (2)
Breit-Wigner distribution:
Γ
1
−iM t − Γt / 2
− Γt
(
)
(
)
.
dS (E) =
a
t
e
p
t
e
→
=
→
=
2
2
2π ( E − M 0 ) + Γ / 4
0
The Breit-Wigner energy distribution cannot be exact.
Two physical conditions for a realistic d S (E )
1) Minimal energy:
2) Mean energy finite:
are:
d S ( E ) = 0 for E < Emin
+∞
+∞
E = ∫-∞ d S ( E ) EdE = ∫E d S ( E ) EdE < ∞
min
Francesco Giacosa
A very simple numerical example
dS (E)
M 0 = 2; Emin = 0.75; Γ = 0.4; Λ = 3
2
2
2
Γ e − (E −E0 )/ Λ θ(E − E min )
dS (E) = N 0
2π (E − M 0 ) 2 + Γ 2 / 4
d BW (E) =
Γ BW
1
2
2π (E − M 0 ) + Γ BW 2 / 4
Γ BW , such that d BW (M 0 ) = dS (M 0 )
+∞
a (t ) = ∫−∞ d S ( E )e −iEt dE ; p (t ) = a (t )
p BW (t ) = e −Γ
BW t
Francesco Giacosa
2
The quantum Zeno effect
We perform N inst. measurements:
the first one at time t = t 0 , the second at time t = 2t 0 , ..., the N-th at time T = Nt 0 .
N


t 02 
T2 
N
pafter N measurements = p(t 0 ) ≈  1 − 2  = 1 − 2 2 
 τZ 
 N τZ 
under the assumption that t 0 is small enough.
If N >> 1 (at fixed T): pafter N measurements ≈ e
−
T2
Nτ2Z
≈ 1.
For large but finite N :
→ slowing down of the decay.
Francesco Giacosa
N
Zeno of Elea
489/431 a.c., Elea
Paradoxes:
Francesco Giacosa
Experimental confirmation of the
quantum Zeno effect - Itano et al (1)
(Undisturbed) survival probability
At t = 0, the electron is in 1 .
Ω2 t 2
 Ωt 
p(t) = cos   = 1 −
+ ...
4
 2 
2
p(T) = 0 für T = π/Ω
Francesco Giacosa
Experimental confirmation of the
quantum Zeno effect - Itano et al (2)
5000 Ions in a Penning trap
Short laser pulses 1-3 work as measurements.
Ω2 t 2
p(t) = cos ( Ωt / 2 ) = 1 −
+ ... ;
4
2
p(T) = 0 für T = π/Ω
(Transition probability (without measuring) at time T) :
1 − p(T) = 1 .
Witn n measurements in between the transition probabilty decreases!
The electron stays in state 1.
Francesco Giacosa
Other experiments about Zeno
Francesco Giacosa
Experimental confirmation of
non-exponential decays (1)
Cold Na atoms in a optical potential
V (x,0)
(
1.0
V ( x, t ) = V0 cos 2k L x − k L at 2
)
0.5
2
4
6
8
x [a.u.]
10
- 0.5
- 1.0
U (x' )
1
x' = x − at 2
2
U ( x' ) = V0 cos(2k L x') + Max'
5
4
3
2
1
x' [a.u.]
2
4
Francesco Giacosa
6
8
10
Experimental confirmation of
non-exponential decays (2)
Measured survival probabilty p(t)
Non-exp decay!
Francesco Giacosa
Experimental confirmation of
non-exponential decays and Zeno /Anti-Zeno effects
Same exp. setup,
but with measurements in between
Zeno effekt
Anti-Zeno effect
Francesco Giacosa
GSI-Anomaly (1)
Measurement of weak decays of ions.
142
61
Pm + e − → υ e +142
Nd
60
ein Zustand
Measurement was:
dN decays
dt
∝−
dp(t)
dt
Oscillations very recently confirmed!
arXiv:1309.7294 [nucl-ex].
Francesco Giacosa
GSI-Anomaly (2)
• Up to now: no explanation of these oscillations!
Neutrino oscillations, (coherent/incoherent sumJ), quantum beats,J
V. P. Krainov, J. of Exp. and Theor. Phys., Vol.115, 68-75
• Simple idea: non-exp. decay due to deviations from the Breit-Wigner limit: Cutoff
h(t) = −
Λ = 32Γ
θ( Λ 2 − (E − M) 2 )
dS (E) = N
(E − M) 2 + Γ 2 / 4
dp
dt
Details in: F. G. and G.Pagliara,
Oscillations in the decay law: A possible quantum mechanical
explanation of the anomaly in the experiment at the GSI facility,
Quant. Matt 2 (2013) 54 [arXiv:1110.1669 [nucl-th]].
Francesco Giacosa
Part 2: Lee Hamiltonian
Francesco Giacosa
Lee Hamiltonian
H = H 0 + H1
+∞
H 0 = M 0 S S + ∫−∞ dkω (k ) k k
H1 =
1
2π
+∞
∫ dk ( g ⋅ f (k ))( S k + k S )
−∞
|S> is the initial unstable state, coupled to an infinity of final states |k>.
(Poincare-time is infinite. Irreversible decay). General approach, similar
Hamiltonians used in many areas of Physics.
Example/1: spontaneous emission. |S> represents an atom in the
excited state, |k> is the ground-state plus photon.
Example/2: pion decay. |S> represents a neutral pion, |k> represents
two photons (flying back-to-back)
Francesco Giacosa
Propagator and spectral function
+∞
H = H 0 + H1 ; H 0 = M 0 S S + ∫ dkω(k) k k ; H1 =
−∞
1
2π
∫
+∞
−∞
dk(g ⋅ f (k))( S k + k S )
−1
G S (E) = S (E − H + iε) S = (E − M 0 + Π (E) + iε)
dS (E) = π1 Im G S (E) ;
a(t) = S e
− iHt
+∞
S = ∫ dEdS (E)e−iEt
−∞
It follows:
∫
+∞
−∞
dEdS (E) = 1
Fermi golden rule: Γ = Im[Π (M)] /2 .
Francesco Giacosa
−1
Π (E) = − ∫
+∞
−∞
dk g 2 f (k) 2
2π E − ω(k) + iε
Exponential limit
+∞
H = H 0 + H1 ; H 0 = M 0 S S + ∫ dkω(k) k k ; H1 =
−∞
1
2π
∫
+∞
−∞
dk(g ⋅ f (k))( S k + k S )
ω(k) = k ; f (k) = 1 ⇒ Π (E) = ig 2 / 2 ; Γ = g 2
1.5
Γ
1
dS (E) =
2π (E − M 0 ) 2 + Γ 2 / 4
⇒ a(t) = e
− i(M0 −iΓ /2)t
⇒ p(t) = e
1.0
0.5
0.0
0
−Γt
Francesco Giacosa
1
2
3
4
5
Exponential limit and final state spectrum (1)
ke
− iHt
2
S
is the prob. that S transforms into k
Translating into energy:
− iω t
− i(M 0 −iΓ /2)t
Γ e −e
η(t, ω) =
2 π E − M 0 + iΓ / 2
2
;
In spont. emission:
η(t, ω)dω is the prob. that the outgoing photon
has an energy between ω and ω+dω
Details in: F. G.,
Energy uncertainty of the final state of a decay process
arXiv:1305.4467 [quant-ph].
Francesco Giacosa
Exponential limit and final state spectrum (2)
η (t , ω ) =
Γ e −e
2π E − M 0 + iΓ / 2
− i ωt
− i ( M 0 − iΓ / 2 ) t
2
Details in: F. G., arXiv:1305.4467 [quant-ph].
Francesco Giacosa
Non-exponential case (1)
H1 =
1
2π
∫
+∞
−∞
 0 for k < E min

f (k) = 1 for E min ≤ k ≤ E max
 0 for k > E
max

dk(g ⋅ f (k))( S k + k S )
dS (E)
1.5
1.0
0.5
0.0
0
1
M 0 = 2; E min = 0; E max
2
3
4
5
E
= 5; g = 0.36 (all in a.u. of energy)
2
Francesco Giacosa
Non-exponential case (2)
p(t)
1.0
0.8
0.6
0.4
0.2
0
2
4
6
Dashed: p BW (t) = e−Γt with Γ = Im[Π (M)] / 2
Francesco Giacosa
8
10
t
Non-exponential case (3)
h(t) = −
dp(t)
dt
Namley: h(t)dt = p(t) − p(t + dt) is the probability that the particles decays between t and t+dt
h(t)
0.35
0.30
0.25
t
∫ h(u)du = 1 − p(t)
0.20
0
0.15
0.10
0.05
2
4
6
Dashed: h BW (t) = Γe−Γt with Γ = Im[Π (M)] / 2
Francesco Giacosa
8
10
t
Two-channel case (1)
H1 =
1
2π
∫
+∞
−∞
dk(g1 ⋅ f1 (k))( S k,1 + k,1 S ) +
1
2π
∫
+∞
−∞
dk(g 2 ⋅ f 2 (k))( S k, 2 + k, 2 S )
1.2

0 for k < E i,min

f i (k) = 1 for E i,min ≤ k ≤ E i,max
 0 for k > E
i,max

1.0
0.8
0.6
0.4
0.2
1
2
3
4
5
M 0 = 2; E1,min = 0; E 2,min = 0 ; E1,max = E 2,max = 5;
g12 = 0.36 ; g 22 = 0.16
Francesco Giacosa
(all in a.u. of energy)
Two-channel case (2)
h1 (t)dt = probabilty that the state S decays in the first channel between (t,t+dt)
h 2 (t)dt = probabilty that the state S decays in the second channel between (t,t+dt)
h1 (t)
h 2 (t)
2.8
2.6
2.4
Dashed:
2.2
h1,BW (t)
h 2,BW (t)
2.0
1.8
2
4
6
8
10
t
Measurable effect???
Details in:
F. G., Non-exponential decay in quantum field theory and in quantum mechanics: the case of two (or more) decay channels,
Found. Phys. 42 (2012) 1262 [arXiv:1110.5923].
Francesco Giacosa
=
Γ1
= const
Γ2
Part 3: Quantum field theory
Francesco Giacosa
Quantum field theory: general properties
Lint = gSϕ2
[g] =[Energy]; QFT super-renorm.
Propagator:
∆S (p 2 ) =
1
p 2 − M 02 + Π (p 2 ) + iε
Spectral function (or energy distribution):
2m
dS (m) =
Im[∆S (p 2 = m 2 )]
π
Normalization follows authomatically:
∫
∞
0
dmdS (m) = 1
F.G. and G. Pagliara, On the spectral functions of scalar mesons,
Phys. Rev. C 76 (2007) 065204 [arXiv:0707.3594].
Francesco Giacosa
Quantum field theory: two examples
Two examples of scalar resonances:
f0(1300) is approx. a relativistic BW resonance
f0(500) is very far from it!!!! (Relevant for chiral theories, nuclear matterJ.)
Francesco Giacosa
Further study of f0(500): position of the pole
F.G. and T. Wolkanowski,
Mod. Phys. Lett. A 27 (2012) 1250229
[arXiv:1209.2332].
Quantum field theory:decay width
Lint = gSϕ2
m2
− µ2
Γtl (m) = 4 2 g 2 ;
4πm
Γtl ( M ) is the tree - level decay width
∞
Γ = ∫ Γtl (m)d S (m)dm
It is an effective inclusion of loop effects!
0
Applications to hadrons (eLSM):
D. Parganlija, F. G. and D. H. Rischke,
Vacuum Properties of Mesons in a Linear Sigma Model with Vector Mesons and Global Chiral Invariance,‚
Phys. Rev. D 82 (2010) 054024 [arXiv:1003.4934 [hep-ph]].
F. Divotgey, L. Olbrich and F. G.,
Phenomenology of axial-vector and pseudovector mesons and their mixing in the kaonic sector,
to appear in EPJA, arXiv:1306.1193 [hep-ph].
J
Francesco Giacosa
Quantum field theory: the decay law
Example: p(t) for the ρ meson
Survival probability amplitude:
∞
a(t) = ∫ dmdS (m)e − imt
0
Just as in QM: non-trivial result!
No dep. on cutoff for a
superrenormalizable field theory
Details in: F. G. and G. Pagliara,
Deviation from the exponential decay law in relativistic quantum field theory: the example of strongly decaying particles,
Mod. Phys. Lett. A 26 (2011) 2247 [arXiv:1005.4817 [hep-ph]].
Francesco Giacosa
Quantum field theory: two-channel case
Lint = g1Sϕ12 + g 2Sϕ22
h1 (t)dt = probabilty that the state S decays in the first channel between (t,t+dt)
h 2 (t)dt = probabilty that the state S decays in the second channel between (t,t+dt)
h1 (t)
h 2 (t)
Details in: F. Giacosa,
Non-exponential decay in quantum field theory and in quantum mechanics: the case of two (or more) decay channels,'
Found. Phys. 42 (2012) 1262 [arXiv:1110.5923 [nucl-th]].
Francesco Giacosa
Quantum field theory: can we „see“ the maximal
energy scale? (1)
Infinities, renormalization, high energy scale,J
Lint = gHψ ψ
This is a renorm. theory.
Calculation of the energy distribution d H (m)
Francesco Giacosa
Quantum field theory: can we „see“ the maximal
energy scale? (2)
∫
Λ
0
d H (m)dm = 1
d H (m) ∝ 1/ (m ⋅ ln 2 m)
for large m
no matter how large is Λ...
but if one tries to do Λ → ∞ one encounters problems:
normalization, etc.
Finite outcome: even for a renorm. QFT the existence of a maximal energy scale
(i.e., a minimal length) is needed.
F. G. and G.Pagliara, Spectral function of a scalar boson coupled to fermions, Phys. Rev. D 88 (2013) 025010 [arXiv:1210.4192].
Francesco Giacosa
Summary and outlook
•
The decay is never exponential! This is a fact.
•
QM: Lee Hamiltonian, deviations easily explained;
final state energy spectrum broadens at short t
two-channel case: the ratio!
•
QFT: qualitatively just as in QM!
Deviations from exp. in particle physics.
Two-channel decay also here interesting.
Minimal length scale.
Francesco Giacosa
Thank You!
Francesco Giacosa