Quantum Mechanics and Quantum Field Theory of Neutrino Oscillations. Joachim Kopp. Nov 15, 2010, IPNAS Seminar, Virginia Tech.
|
|
- Elfrieda Harmon
- 5 years ago
- Views:
Transcription
1 Quantum Mechanics and Quantum Field Theory of Neutrino Oscillations Joachim Kopp Nov 15, 2010, IPNAS Seminar, Virginia Tech Fermilab based on work done in collaboration with E. Kh. Akhmedov, C. M. Ho, B. Kayser, M. Lindner, R. G. H. Robertson, P. Vogel Joachim Kopp QM and QFT of Neutrino Oscillations 1
2 Outline 1 Motivation 2 Wave packet approaches to neutrino oscillations QM: Neutrino wave packets QFT: Feynman diagrams with external wave packets The connection between wave packet and plane wave approaches 3 Entanglement in neutrino oscillations 4 Neutrino flavor eigenstates 5 Summary and conclusions Joachim Kopp QM and QFT of Neutrino Oscillations 2
3 Outline 1 Motivation 2 Wave packet approaches to neutrino oscillations QM: Neutrino wave packets QFT: Feynman diagrams with external wave packets The connection between wave packet and plane wave approaches 3 Entanglement in neutrino oscillations 4 Neutrino flavor eigenstates 5 Summary and conclusions Joachim Kopp QM and QFT of Neutrino Oscillations 3
4 The textbook approach to neutrino oscillations Diagonalization of the mass terms of the charged leptons and neutrinos gives L g 2 (ē αl γ µ U αj ν jl ) W µ + diag. mass terms + h.c. (flavour eigenstates: α = e, µ, τ, mass eigenstates: j = 1, 2, 3) Assume, at time t = 0 and location x = 0, a flavour eigenstate ν(0, 0) = ν α = i U αj ν j is produced. At time t and position x, it has evolved into ν(t, x) = i U αje ie j t+i p j x ν i Oscillation probability: P(ν α ν β ) = ν β ν(t, x) 2 = UαjU βj U αk Uβke i(e j E k )t+i( p j p k ) x j,k Joachim Kopp QM and QFT of Neutrino Oscillations 4
5 Questions to think about What happens to energy-momentum conservation in neutrino oscillations? Under what conditions is coherence between different neutrino mass eigenstates lost? Is the standard oscillation formula universal, or are there situations where it needs to be modified? Why does the standard oscillation formula work so well, even though its derivation is plagued by inconsistencies? Are the neutrino and its interaction partners in an entangled state? If so, what are the phenomenological consequences of entanglement? Are flavor eigenstates well-defined and useful objects in QFT? Joachim Kopp QM and QFT of Neutrino Oscillations 5
6 Outline 1 Motivation 2 Wave packet approaches to neutrino oscillations QM: Neutrino wave packets QFT: Feynman diagrams with external wave packets The connection between wave packet and plane wave approaches 3 Entanglement in neutrino oscillations 4 Neutrino flavor eigenstates 5 Summary and conclusions Joachim Kopp QM and QFT of Neutrino Oscillations 6
7 Neutrino wave packets Treat neutrino as superposition of mass eigenstate wave packets: ν αs (t) = d 3 p Uαj f js ( p) e ie j ( p)(t t S ) ν j, p j Advantage: Heisenberg uncertainties and localization effects properly taken into account. Disadvatage: Assumptions on (or calculation of) shape functions f js ( p) needed. Joachim Kopp QM and QFT of Neutrino Oscillations 7
8 Oscillation formula for neutrino wave packets Detector projects ν αs (t) onto a state ν βd with shape function f kd ( p). Oscillation probability (for Gaussian wave packets): P ee = j,k [ UαjU βj U αk Uβk exp 2πi L L osc jk ( ) 2 ( ) 2 ] L 1 L coh 2π 2 ξ 2 jk 2σ p L osc jk with C. Giunti, C. W. Kim, U. W. Lee, Phys. Rev. D44 (1991) 3635; C. Giunti, C. W. Kim, Phys. Lett. B274 (1992) 87 K. Kiers, S. Nussinov, N. Weiss, Phys. Rev. D53 (1996) 537, hep-ph/ C. Giunti, C. W. Kim, Phys. Rev. D58 (1998) , hep-ph/ , C. Giunti, Found. Phys. Lett. 17 (2004) 103, hep-ph/ L coh L osc jk = 4πE/ m 2 jk Oscillation length jk = 2 2E 2 /σ p mjk 2 Coherence length E Energy that a massless neutrino would have ξ quantifies the deviation of E i from E Effective wave packet width σ p Joachim Kopp QM and QFT of Neutrino Oscillations 8
9 The localization term For ξ 1, the term [ ( ) 2 ] 1 exp 2π 2 ξ 2 2σ p L osc jk suppresses oscillations if wave packets are larger than the oscillation length. In that case, the phase difference between mass eigenstates varies appreciably during the detection process. For ξ 1 (equal energy limit, E i E j ), this condition disappears (oscillations only in L, not in T phase relation between mass eigenstates remains the same during the whole detection process). Joachim Kopp QM and QFT of Neutrino Oscillations 9
10 The decoherence term The term exp [ ( ) 2 ] L L coh jk suppresses oscillations at long (astrophysical) baselines. Interesting duality of interpretations Interpretation 1: (a single neutrino) Different mass eigenstates separate in space and time due to their different group velocities loss of coherence Interpretation 2: (ensemble of neutrinos) At large baseline, oscillation maxima are so close in energy that the finite experimental resolution smears them out. Reason: A continuous flux of identical wave packets cannot be distinguished from an ensemble of plane waves with the same momentum distribution. (Neutrino density matrices for the two ensembles are identical) K. Kiers, S. Nussinov, N. Weiss, Phys. Rev. D53 (1996) 537, hep-ph/ Joachim Kopp QM and QFT of Neutrino Oscillations 10
11 The Feynman diagram of neutrino oscillations Idea: Neutrino as intermediate line in macroscopic Feynman diagram; external particles described as wave packets or bound state wave functions P f(k) D f(k ) ν P i(q) Advantages: Heisenberg uncertainties and localization effects properly taken into account. Neutrino properties (E j, p j of individual mass eigenstates, shape of wave function,... ) automatically determined by the formalism. Automatically yields properly normalized oscillation probabilities and event rates Includes dynamic modifications of mixing parameters due to different kinematics for different mass eigenstates (usually tiny, but may be important in sterile neutrino scenarios) D i(q ) Joachim Kopp QM and QFT of Neutrino Oscillations 11
12 Oscillation probability from a Feynman diagram Evaluation is tedious, but straightforward if external wave functions are Gaussian: (δm 2 i = m 2 i m 2 0 ) P αβ (L) j,k [ UαjU αk UβkU βj exp 2πi L L osc jk ( ) 2 L L coh jk ( ) 2 π2 1 2 ξ2 σ p L osc (δm2 i + δmj 2)2 jk 32σmE 2 2 (δm2 i δmj 2)2 32σmE 2 2 ], Four terms: Oscillation Decoherence (wave packet separation detector resolution effects) Localization (wave packet smaller than oscillation length or no oscillations in time) Approximate conservation of average energies/momenta Another localization effect: Suppression of oscillations if experimental resolutions allow determination of neutrino mass B. Kayser Phys. Rev. D24 (1981) 110 Joachim Kopp QM and QFT of Neutrino Oscillations 12
13 A concrete example: Mössbauer neutrinos Classical Mössbauer effect: Recoilfree emission and absorption of γ-rays from nuclei bound in a crystal lattice. A similar effect should exist for neutrino emission/absorption in bound state β decay and induced electron capture processes. Proposed experiment: W. M. Visscher, Phys. Rev. 116 (1959) 1581; W. P. Kells, J. P. Schiffer, Phys. Rev. C28 (1983) 2162 R. S. Raghavan, hep-ph/ ; R. S. Raghavan, hep-ph/ Production: Detection: 3 H 3 He + + ν e + e (bound) 3 He + + e (bound) + ν e 3 H 3 H and 3 He embedded in metal crystals (metal hydrides). Physics opportunities: Neutrino oscillations on a laboratory scale: E = 18.6 kev, L osc atm 20 m. Gravitational interactions of neutrinos Study of solid state effects with unprecedented precision Joachim Kopp QM and QFT of Neutrino Oscillations 13
14 Mössbauer neutrinos (contd.) Mössbauer neutrinos have very special properties: Neutrino receives full decay energy: Q = 18.6 kev Natural line width: γ ev Atucal line width: γ ev Inhomogeneous broadening (Impurities, lattice defects) Each atom emits at slightly different, but constant, energy ensemble of plane wave with different energies Homogeneous broadening (Spin interactions) Energy levels of atoms fluctuate, but in the same way for all atoms ensemble of wave packets with identical shapes R. S. Raghavan, W. Potzel Remember: A continuous flux of identical wave packets cannot be distinguished from an ensemble of plane waves with the same momentum distribution. (Neutrino density matrices for the two ensembles are identical) K. Kiers, S. Nussinov, N. Weiss, Phys. Rev. D53 (1996) 537, hep-ph/ Joachim Kopp QM and QFT of Neutrino Oscillations 14
15 Transition amplitude for Mössbauer neutrinos Assume atoms bound in harmonic oscillator potentials, e.g. for 3 H in the source: [ mh ω H,S ψ H,S ( x, t) = π ] 3 4 exp [ 1 2 m Hω H,S x x S 2 ] e ie H,St Joachim Kopp QM and QFT of Neutrino Oscillations 15
16 Transition amplitude for Mössbauer neutrinos (contd.) Z Z ia = d 3 x 1 dt 1 d 3 mh ω H,S x 2 dt 2 π mhe ω He,S π mhe ω He,D π mh ω H,D π X Z M µ M ν U ej 2 j ū e,s γ µ(1 γ 5 ) «3 4 exp» 12 m Hω H,S x 1 x S 2 e ie H,S t 1 «3 4 exp» 12 m Heω He,S x 1 x S 2 e +ie He,S t 1 «3 4 exp» 12 m Heω He,D x 2 x D 2 e ie He,Dt 2 «3 4 exp» 12 m Hω H,D x 2 x D 2 e +ie H,Dt 2 d 4 p (2π) 4 e ip 0(t 2 t 1 )+i p( x 2 x 1 ) i(/p + m j ) p 2 0 p2 m 2 j + iɛ (1 + γ5 )γ νu e,d. dt 1 dt 2 -integrals energy-conserving δ functions p 0 -integral trivial d 3 x 1 d 3 x 2 -integrals are Gaussian d 3 p-integral: Use Grimus-Stockinger theorem (limit of propagator for large L = x D x S ). W. Grimus, P. Stockinger, Phys. Rev. D54 (1996) 3414, hep-ph/ Joachim Kopp QM and QFT of Neutrino Oscillations 16
17 From the amplitude to the transition rate Amplitude: ia = i [ 2L N δ(e S E D ) exp E S 2 ] m2 j q 2σp 2 M µ M ν U ej 2 e i E 2 S m2 L j j σ 2 p = (m H ω H,S + m He ω He,S ) 1 + (m H ω H,D + m He ω He,D ) 1 ū e,s γ µ 1 γ 5 2 (/p j + m j ) 1+γ5 2 γ ν u e,d, Joachim Kopp QM and QFT of Neutrino Oscillations 17
18 From the amplitude to the transition rate Amplitude: ia = i [ 2L N δ(e S E D ) exp E S 2 ] m2 j q 2σp 2 M µ M ν U ej 2 e i E 2 S m2 L j j σ 2 p = (m H ω H,S + m He ω He,S ) 1 + (m H ω H,D + m He ω He,D ) 1 ū e,s γ µ 1 γ 5 2 (/p j + m j ) 1+γ5 2 γ ν u e,d, Transition rate: Integrate A 2 over densities of initial and final states Γ 0 de H,S de He,S de He,D de H,D δ(e S E D )ρ H,S (E H,S ) ρ He,D (E He,D ) ρ He,S (E He,S ) ρ H,D (E H,D ) [ U ej 2 U ek 2 exp 2E S 2 m2 j mk 2 ] (q e i E 2 S m2 E j 2 S m2 k j,k 2σ 2 p }{{} Analogue of Lamb-Mössbauer factor (Probability of recoil-free transition) ) L } {{ } Oscillation phase Joachim Kopp QM and QFT of Neutrino Oscillations 17
19 Inhomogeneous line broadening Energy levels of 3 H and 3 He in the source and detector are smeared e.g. due to crystal impurities, lattice defects, etc. R. S. Raghavan, hep-ph/ , W. Potzel, Phys. Scripta T127 (2006) 85, B. Balko, I. W. Kay, J. Nicoll, J. D. Silk, G. Herling, Hyperfine Int. 107 (1997) 283 Good approximation for distribution of energy levels: ρ A,B (E A,B ) = γ A,B /2π (E A,B E A,B,0 ) 2 + γ 2 A,B /4 Mössbauer neutrino transition rate for two neutrino flavours (m 2 > m 1 ): [ Γ exp E S,0 2 ] [ ] m2 2 σp 2 exp m2 (γ S + γ D )/2π 2σp 2 (E S,0 E D,0 ) 2 + (γ S+γ D ) 2 4 [ {1 2s 2 c ( 2 (e L/Lcoh S + e L/Lcoh D )cos π L )]} L osc Oscillation term Lamb-Mössbauer factor Breit-Wigner resonance term Coherence terms with L coh S,D = 4Ē 2 / m 2 γ S,D ( L osc in reality) Note: No localization term exp[ 2π 2 ξ 2 (1/2σ p L osc jk )2 ] since E i E j. Joachim Kopp QM and QFT of Neutrino Oscillations 18
20 Homogeneous line broadening Fluctuating electromagnetic fields in solid state crystal Fluctuating energy levels of 3 H and 3 He. Classical Mössbauer effect: Homogeneous and inhomogeneous broadening both lead to Lorentzian line shapes Experimentally indistinguishable We expect a result similar to that for the case of inhomogeneous broadening (Rember Kiers, Nussinov, Weiss!) Ansatz: Introduce modulation factors of the form [ t f A,B (t) = exp i dt [ ] E A,B (t ] ) E A,B,0 t 0 in the 3 H and 3 He wave functions (A = H, He, B = S, D). J. Odeurs, Phys. Rev. B52 (1995) 6166 Joachim Kopp QM and QFT of Neutrino Oscillations 19
21 Transition amplitude for homogeneous line broadening Z Z «3 ia = d 3 x 1 dt 1 d 3 mh ω 4 H,S x 2 dt 2 exp» 12 π m Hω H,S x 1 x S 2 «3 mhe ω 4 He,S exp» 12 π m Heω He,S x 1 x S 2 e +ie He,S t 1 «3 mhe ω 4 He,D exp» 12 π m Heω He,D x 2 x D 2 e ie He,Dt 2 «3 mh ω 4 H,D exp» 12 π m Hω H,D x 2 x D 2 e +ie H,Dt 2 X Z M µ S Mν D U ej 2 d 4 p (2π) exp ˆ ip 4 0 (t 2 t 1 ) + i p( x 2 x 1 ) j ū e,s γ µ(1 γ 5 i(/p + m j ) ) p0 2 p2 mj 2 + iɛ (1 + γ5 )γ νu e,d e ie H,S t 1 Joachim Kopp QM and QFT of Neutrino Oscillations 20
22 Transition amplitude for homogeneous line broadening Z Z ia = d 3 x 1 dt 1 d 3 mh ω H,S x 2 dt 2 π mhe ω He,S π mhe ω He,D π mh ω H,D π X Z M µ S Mν D U ej 2 j «3 4 exp» 12 m Hω H,S x 1 x S 2 f H,S (t 1 ) e ie H,S t 1 «3 4 exp» 12 m Heω He,S x 1 x S 2 f He,S(t 1 ) e +ie He,S t 1 «3 4 exp» 12 m Heω He,D x 2 x D 2 f He,D (t 2 ) e ie He,Dt 2 «3 4 exp» 12 m Hω H,D x 2 x D 2 f H,D(t 2 ) e +ie H,Dt 2 d 4 p (2π) exp ˆ ip 4 0 (t 2 t 1 ) + i p( x 2 x 1 ) ū e,s γ µ(1 γ 5 i(/p + m j ) ) p0 2 p2 mj 2 + iɛ (1 + γ5 )γ νu e,d Joachim Kopp QM and QFT of Neutrino Oscillations 20
23 Transition amplitude for homogeneous line broadening Z Z ia = d 3 x 1 dt 1 d 3 mh ω H,S x 2 dt 2 π mhe ω He,S π mhe ω He,D π mh ω H,D π X Z M µ S Mν D U ej 2 j «3 4 exp» 12 m Hω H,S x 1 x S 2 f H,S (t 1 ) e ie H,S t 1 «3 4 exp» 12 m Heω He,S x 1 x S 2 f He,S(t 1 ) e +ie He,S t 1 «3 4 exp» 12 m Heω He,D x 2 x D 2 f He,D (t 2 ) e ie He,Dt 2 «3 4 exp» 12 m Hω H,D x 2 x D 2 f H,D(t 2 ) e +ie H,Dt 2 d 4 p (2π) exp ˆ ip 4 0 (t 2 t 1 ) + i p( x 2 x 1 ) ū e,s γ µ(1 γ 5 i(/p + m j ) ) p0 2 p2 mj 2 + iɛ (1 + γ5 )γ νu e,d d 3 x 1 d 3 x 2 -integrals are Gaussian d 3 p-integral: Use Grimus-Stockinger theorem. Transition rate Γ AA (average of AA over all possible 3 H and 3 He states) can be computed if fluctuations are Markovian (system has no memory) Joachim Kopp QM and QFT of Neutrino Oscillations 20
24 Transition rate for homogeneous line broadening Result: [ Γ exp E S,0 2 ] [ ] m2 2 σp 2 exp m2 (γ S + γ D )/2π 2σp 2 (E S,0 E D,0 ) 2 + (γ S+γ D ) 2 4 [ {1 2s 2 c ( 2 (e L/Lcoh S + e L/Lcoh D )cos π L )]} L osc... identical to the result for inhomogeneous line broadening. This illustrates that... A continuous flux of identical wave packets cannot be distinguished from an ensemble of plane waves with the same momentum distribution. (Neutrino density matrices for the two ensembles are identical) K. Kiers, S. Nussinov, N. Weiss, Phys. Rev. D53 (1996) 537, hep-ph/ Joachim Kopp QM and QFT of Neutrino Oscillations 21
25 Plane waves vs. wave packets We know: Plane wave calculations yield the correct oscillation formula even though they are plagued by inconsistencies Plane waves are infinitely delocalized, but concept of baseline requires localization Relation L T (required in the derivation of the oscillation probability unless E i E j ) cannot be justified. Perfect knowledge of E j and p j means we know m j no oscillations B. Kayser Phys. Rev. D24 (1981) So, why do plane wave approaches work so well?!? Joachim Kopp QM and QFT of Neutrino Oscillations 22
26 Reason 1: Statistical ensembles of neutrinos A continuous flux of identical wave packets cannot be distinguished from an ensemble of plane waves with the same momentum distribution. (Neutrino density matrices for the two ensembles are identical) K. Kiers, S. Nussinov, N. Weiss, Phys. Rev. D53 (1996) 537, hep-ph/ For a time-independent flux of neutrinos: Wave packet picture equivalent to plane wave picture Production and detection can still be localized, i.e. density matrix ρ(x) is probed at fixed x = L. Density matrix is diagonal in only states with same energy (E i = E j ) can interfere no T -dependence, relation L T not required. No perfect knowledge of E j and p j for each individual neutrino oscillations still possible Joachim Kopp QM and QFT of Neutrino Oscillations 23
27 Illustration L. Stodolsky, hep-ph/ If time-of-flight information is not available, only energy eigenstates interfere. Consider neutrino density matrix ρ ψ n ψ n many neutrinos von Neumann equation: ρ = i[h, ρ] Stationary (time-independent) neutrino flux: ρ = 0 ρ and H commute; ρ can be written as ensemble of energy eigenstates ρ = de c(e) E E ( ) If ρ 0, but time information is discarded ( dt ρ), energy-off-diagonal elements (containg factors e i(e i E j )t ) will be averaged to zero. same situation as for stationary flux, ( ) holds. Joachim Kopp QM and QFT of Neutrino Oscillations 24
28 Reason 2: Factorizing out wave packet effects Consider Gaussian wave packet: ψ α ( x, t) j [ Uαje ie j0t+i p x j0 ( x v j t) 2 ] exp Plane wave, multiplied with shape factor At a given spacetime point, phase difference depends only on average energies E j0 and momenta p j0. Oscillation length cannot depend on wave packet shape and width. (Only wave packet components located at the same spacetime point can interfere.) 4σ 2 x The wave packet approach shows that the assumptions made in the plane wave picture are justified and consistent. Joachim Kopp QM and QFT of Neutrino Oscillations 25
29 Outline 1 Motivation 2 Wave packet approaches to neutrino oscillations QM: Neutrino wave packets QFT: Feynman diagrams with external wave packets The connection between wave packet and plane wave approaches 3 Entanglement in neutrino oscillations 4 Neutrino flavor eigenstates 5 Summary and conclusions Joachim Kopp QM and QFT of Neutrino Oscillations 26
30 Entanglement in neutrino oscillations Consider two-body decay π µ + ν µ. Question: Are ν µ and µ kinematically entangled? Should the final state wave function be written as Uµj µ j (P j, X) ν j (p j, x)? j YES! Without observation, all possible final states are produced simultaneously. Entanglement used successfully in b physics NO! Energy-momentum uncertainty implies that every p can come with many different P (wave packets!). Only Feynman diagrams with identical external legs can interfere. Cohen Glashow Ligeti; Akhmedov Smirnov; Kayser Kopp Robertson Vogel Joachim Kopp QM and QFT of Neutrino Oscillations 27
31 The entangled point of view Describe particles as plane waves with precisely known 4-momenta: π(p π ) µ(p j ) + ν j (p j ) where p i p j, P i P j due to E p conservation. Energy and momentum uncertainties in the production/detection processes imply that there is a spectrum of p π, p i, P i. Uncertainties allow interference between states with different p i, P i. The contribution of µ(p j ) to the phase of the entangled state is the same for all P j : Akhmedov Smirnov [ E(Pi ) E(P j ) ] T [ P i P j ] L = [ E 0 + δe i E 0 δe j ] T [ E0 + δe i E 0 P 0 E 0 δe j E 0 P 0 ] L = [ δe i δe j ]( T L/v ) 0 the charged lepton does not oscillate. Joachim Kopp QM and QFT of Neutrino Oscillations 28
32 The wavepacket point of view P f(k) D f(k ) ν P i(q) D i(q ) External particles (parent pion, nucleus in the detector,... ) are observed their state at t ± is fixed by the observation Each neutrino mass eigenstate may interact with different momentum components of the external particles wave packets. (in a sense, there is entanglement between the neutrino and components of the wave packet. This is automatically taken care of by the E p conserving δ-functions at the Feynman vertices.) Joachim Kopp QM and QFT of Neutrino Oscillations 29
33 Outline 1 Motivation 2 Wave packet approaches to neutrino oscillations QM: Neutrino wave packets QFT: Feynman diagrams with external wave packets The connection between wave packet and plane wave approaches 3 Entanglement in neutrino oscillations 4 Neutrino flavor eigenstates 5 Summary and conclusions Joachim Kopp QM and QFT of Neutrino Oscillations 30
34 Neutrino flavor eigenstates Flavor fields defined as ν α = U αj ν j. The problem: Flavor f α does not commute with the Hamiltonian. (for E i E j ) [ [ ] ] F, H f α ν α ν α, E k ν k ν k α=e,µ,τ = α=e,µ,τ = α=e,µ,τ f α [ f α i,j=1,2,3 i,j=1,2,3 k=1,2,3 U αiu αj ν i ν j, k=1,2,3 ] E k ν k ν k ) UαiU αj (E j ν i ν j E i ν j ν i Therefore, flavor eigenstates are not meaningful tools for calculating transition amplitudes. Joachim Kopp QM and QFT of Neutrino Oscillations 31
35 Outline 1 Motivation 2 Wave packet approaches to neutrino oscillations QM: Neutrino wave packets QFT: Feynman diagrams with external wave packets The connection between wave packet and plane wave approaches 3 Entanglement in neutrino oscillations 4 Neutrino flavor eigenstates 5 Summary and conclusions Joachim Kopp QM and QFT of Neutrino Oscillations 32
36 Answers What happens to energy-momentum conservation in neutrino oscillations? In the wave packet approach, average momenta need not be conserved. For each momentum component of the wave packets, E p conservation holds. No energy-momentum non-conservation beyond what is allowed by the Heisenberg uncertainties. If coherent superposition of neutrino mass eigenstates is forbidden by E p conservation (hypothetical experiment with excellent E and p resolutions), oscillations vanish (ensured by localization terms). Under what conditions is coherence between different neutrino mass eigenstates lost? For L L coh, wave packets separate loss of coherence (equivalently: Poor energy resolution smears out oscillations) No coherence if experiment allows determination of neutrino mass Is the standard oscillation formula universal, or are there situations where it needs to be modified? No situation with observable deviations has been found. Mössbauer neutrinos oscillate in the standard way. Entanglement does not lead to new phenomena. Joachim Kopp QM and QFT of Neutrino Oscillations 33
37 Answers (contd.) Why does the standard oscillation formula work so well, even though its derivation is plagued by inconsistencies? Wave packet effects can be factorized out. Statistical ensemble of wave packets equivalent to statistical ensemble of plane waves. No inconsistencies in this case. Are the neutrino and its interaction partners in an entangled state? If so, what are the phenomenological consequences of entanglement? Entanglement is a valid, but unnecessary concept. No phenomenological consequences of entanglement. Are flavor eigenstates well-defined and useful objects in QFT? Flavor eigenstates can be defined, but are useless for matrix element calculations because they do not commute with H. Joachim Kopp QM and QFT of Neutrino Oscillations 34
38 Thank you!
Mössbauer effect with neutrinos and neutrino oscillations
Mössbauer effect with neutrinos and neutrino oscillations Based on: EA, J. Kopp & M. Lindner, JHEP 0805:005,2008 [arxiv:0802.2513] and arxiv:0803.1424 Evgeny Akhmedov MPI-K, Heidelberg Evgeny Akhmedov
More informationNeutrino Oscillations from Quantum Field Theory
Neutrino Oscillations from Quantum Field Theory Partly based on: EA, J. Kopp & M. Lindner, JHEP 0805:005,2008 [arxiv:0802.2513] and EA & J. Kopp, work in progress Evgeny Akhmedov MPI-K, Heidelberg & Kurchatov
More informationNeutrino Oscillations through Entanglement
eutrino Oscillations through Entanglement Tess E. Smidt (Dated: June 10, 2011) Despite the fact that neutrino oscillations have been theoretically and experimentally studied over the past half-century,
More informationThe Physics of Neutrino Oscillation
The Physics of Neutrino Oscillation Boris Kayser PASI March, 2012 1 Neutrino Flavor Change (Oscillation) in Vacuum + l α (e.g. µ) ( ) Approach of B.K. & Stodolsky - l β (e.g. τ) Amp W (ν α ) (ν β ) ν W
More informationIntroduction to Neutrino Physics. TRAN Minh Tâm
Introduction to Neutrino Physics TRAN Minh Tâm LPHE/IPEP/SB/EPFL This first lecture is a phenomenological introduction to the following lessons which will go into details of the most recent experimental
More informationThe Physics of Neutrino Oscillation. Boris Kayser INSS August, 2013
The Physics of Neutrino Oscillation Boris Kayser INSS August, 2013 1 Neutrino Flavor Change (Oscillation) in + l α (e.g. µ) Vacuum ( Approach of ) B.K. & Stodolsky - l β (e.g. τ) Amp ν W (ν α ) (ν β )
More informationWave Packet Description of Neutrino Oscillation in a Progenitor Star Supernova Environment
Brazilian Journal of Physics, vol. 37, no. 4, December, 2007 1273 Wave Packet Description of Neutrino Oscillation in a Progenitor Star Supernova Environment F. R. Torres and M. M. Guzzo Instituto de Física
More informationDamping signatures in future neutrino oscillation experiments
Damping signatures in future neutrino oscillation experiments Based on JHEP 06(2005)049 In collaboration with Tommy Ohlsson and Walter Winter Mattias Blennow Division of Mathematical Physics Department
More informationRelativistic Quantum Theories and Neutrino Oscillations. Abstract
Relativistic Quantum Theories and Neutrino Oscillations B. D. Keister Physics Division, 1015N National Science Foundation 4201 Wilson Blvd. Arlington, VA 22230 W. N. Polyzou Department of Physics and Astronomy
More informationTheory of Neutrino Oscillations
INCONTRI SULLA FISICA DELLE ALTE ENERGIE TORINO, APRIL 14-16, 2004 Theory of Neutrino Oscillations Carlo Giunti 1) arxiv:hep-ph/0409230v1 20 Sep 2004 1. INFN, Sezione di Torino, and Dipartimento di Fisica
More informationNeutrinos: Three-Flavor Effects in Sparse and Dense Matter
Neutrinos: Three-Flavor Effects in Sparse and Dense Matter Tommy Ohlsson tommy@theophys.kth.se Royal Institute of Technology (KTH) & Royal Swedish Academy of Sciences (KVA) Stockholm, Sweden Neutrinos
More informationNeutrino Phenomenology. Boris Kayser ISAPP July, 2011 Part 1
Neutrino Phenomenology Boris Kayser ISAPP July, 2011 Part 1 1 What Are Neutrinos Good For? Energy generation in the sun starts with the reaction Spin: p + p "d + e + +# 1 2 1 2 1 1 2 1 2 Without the neutrino,
More informationNeutrino Oscillations And Sterile Neutrinos
Neutrino Oscillations And Sterile Neutrinos Keshava Prasad Gubbi University of Bonn s6nagubb@uni-bonn.de May 27, 2016 Keshava Prasad Gubbi (Uni Bonn) Neutrino Oscillations,Sterile Neutrinos May 27, 2016
More informationNEUTRINO PHYSICS. Neutrino Oscilla,on Theory. Fabio Bellini. Fisica delle Par,celle Elementari, Anno Accademico Lecture november 2013
NEUTRINO PHYSICS Neutrino Oscilla,on Theory Lecture 16 11 november 2013 Fabio Bellini Fisica delle Par,celle Elementari, Anno Accademico 2013-2014 http://www.roma1.infn.it/people/rahatlou/particelle 2
More informationPhys 622 Problems Chapter 5
1 Phys 622 Problems Chapter 5 Problem 1 The correct basis set of perturbation theory Consider the relativistic correction to the electron-nucleus interaction H LS = α L S, also known as the spin-orbit
More informationMass hierarchy determination in reactor antineutrino experiments at intermediate distances. Promises and challenges. Petr Vogel, Caltech
Mass hierarchy determination in reactor antineutrino experiments at intermediate distances. Promises and challenges. Petr Vogel, Caltech Mass hierarchy ν e FLAVOR FLAVOR ν µ ν τ ν 3 ν 2 ν 1 m 2 21 MASS
More informationNeutrino wave function and oscillation suppression.
Neutrino wave function and oscillation suppression. A.D. Dolgov a,b, O.V. Lychkovskiy, c, A.A. Mamonov, c, L.B. Okun a, and M.G. Schepkin a a Institute of Theoretical and Experimental Physics 11718, B.Cheremushkinskaya
More informationWhat We Know, and What We Would Like To Find Out. Boris Kayser Minnesota October 23,
What We Know, and What We Would Like To Find Out Boris Kayser Minnesota October 23, 2008 1 In the last decade, observations of neutrino oscillation have established that Neutrinos have nonzero masses and
More informationOn the group velocity of oscillating neutrino states and the equal velocities assumption
On the group velocity of oscillating neutrino states and the equal velocities assumption J.-M. Lévy September 6, 2003 Abstract In a two flavour world, the usual equal p, equale, orequalv assumptions used
More informationWeak interactions. Chapter 7
Chapter 7 Weak interactions As already discussed, weak interactions are responsible for many processes which involve the transformation of particles from one type to another. Weak interactions cause nuclear
More informationWave-packet treatment of neutrino oscillations and its implication on Daya Bay and JUNO experiments
Wave-packet treatment of neutrino oscillations and its implication on Daya Bay and JUNO experiments Steven C.F.Wong Sun Yat-Sen University 22nd May 2015 Overview Part I: My past and current researches
More informationOverview of mass hierarchy, CP violation and leptogenesis.
Overview of mass hierarchy, CP violation and leptogenesis. (Theory and Phenomenology) Walter Winter DESY International Workshop on Next Generation Nucleon Decay and Neutrino Detectors (NNN 2016) 3-5 November
More informationQuantum Electronics/Laser Physics Chapter 4 Line Shapes and Line Widths
Quantum Electronics/Laser Physics Chapter 4 Line Shapes and Line Widths 4.1 The Natural Line Shape 4.2 Collisional Broadening 4.3 Doppler Broadening 4.4 Einstein Treatment of Stimulated Processes Width
More informationNeutrino Oscillations and the Matter Effect
Master of Science Examination Neutrino Oscillations and the Matter Effect RAJARSHI DAS Committee Walter Toki, Robert Wilson, Carmen Menoni Overview Introduction to Neutrinos Two Generation Mixing and Oscillation
More informationS.K. Saikin May 22, Lecture 13
S.K. Saikin May, 007 13 Decoherence I Lecture 13 A physical qubit is never isolated from its environment completely. As a trivial example, as in the case of a solid state qubit implementation, the physical
More informationNeutrino Phenomenology. Boris Kayser INSS August, 2013 Part 1
Neutrino Phenomenology Boris Kayser INSS August, 2013 Part 1 1 What Are Neutrinos Good For? Energy generation in the sun starts with the reaction Spin: p + p "d + e + +# 1 2 1 2 1 1 2 1 2 Without the neutrino,
More informationarxiv: v1 [quant-ph] 11 Nov 2014
Electric dipoles on the Bloch sphere arxiv:1411.5381v1 [quant-ph] 11 Nov 014 Amar C. Vutha Dept. of Physics & Astronomy, York Univerity, Toronto ON M3J 1P3, Canada email: avutha@yorku.ca Abstract The time
More informationUNIVERSITY OF SOUTHAMPTON
UNIVERSITY OF SOUTHAMPTON PHYS6012W1 SEMESTER 1 EXAMINATION 2012/13 Coherent Light, Coherent Matter Duration: 120 MINS Answer all questions in Section A and only two questions in Section B. Section A carries
More informationarxiv:hep-ph/ v2 3 Apr 2003
3 April 2003 hep-ph/0302026 Coherence and Wave Packets in Neutrino Oscillations C. Giunti arxiv:hep-ph/0302026v2 3 Apr 2003 INFN, Sezione di Torino, and Dipartimento di Fisica Teorica, Università di Torino,
More informationLikewise, any operator, including the most generic Hamiltonian, can be written in this basis as H11 H
Finite Dimensional systems/ilbert space Finite dimensional systems form an important sub-class of degrees of freedom in the physical world To begin with, they describe angular momenta with fixed modulus
More informationCoherent states, beam splitters and photons
Coherent states, beam splitters and photons S.J. van Enk 1. Each mode of the electromagnetic (radiation) field with frequency ω is described mathematically by a 1D harmonic oscillator with frequency ω.
More informationThe GSI Anomaly. M. Lindner. Max-Planck-Institut für Kernphysik, Heidelberg. Sildes partially adopted from F. Bosch
The GSI Anomaly M. Lindner Max-Planck-Institut für Kernphysik, Heidelberg Sildes partially adopted from F. Bosch What is the GSI Anomaly? Periodically modualted exponential β-decay law of highly charged,
More informationInteractions/Weak Force/Leptons
Interactions/Weak Force/Leptons Quantum Picture of Interactions Yukawa Theory Boson Propagator Feynman Diagrams Electromagnetic Interactions Renormalization and Gauge Invariance Weak and Electroweak Interactions
More informationSolar neutrinos and the MSW effect
Chapter 12 Solar neutrinos and the MSW effect The vacuum neutrino oscillations described in the previous section could in principle account for the depressed flux of solar neutrinos detected on Earth.
More informationA Simple Model of Quantum Trajectories. Todd A. Brun University of Southern California
A Simple Model of Quantum Trajectories Todd A. Brun University of Southern California Outline 1. Review projective and generalized measurements. 2. A simple model of indirect measurement. 3. Weak measurements--jump-like
More informationarxiv: v3 [hep-ph] 23 Jan 2017
Effects of Matter in Neutrino Oscillations and Determination of Neutrino Mass Hierarchy at Long-baseline Experiments T. Nosek Institute of Particle and Nuclear Physics, Faculty of Mathematics and Physics,
More informationInteractions/Weak Force/Leptons
Interactions/Weak Force/Leptons Quantum Picture of Interactions Yukawa Theory Boson Propagator Feynman Diagrams Electromagnetic Interactions Renormalization and Gauge Invariance Weak and Electroweak Interactions
More informationQuestioning Quantum Mechanics? Kurt Barry SASS Talk January 25 th, 2012
Questioning Quantum Mechanics? Kurt Barry SASS Talk January 25 th, 2012 2 Model of the Universe Fundamental Theory Low-Energy Limit Effective Field Theory Quantum Mechanics Quantum Mechanics is presently
More informationNeutrino Anomalies & CEνNS
Neutrino Anomalies & CEνNS André de Gouvêa University PIRE Workshop, COFI February 6 7, 2017 Something Funny Happened on the Way to the 21st Century ν Flavor Oscillations Neutrino oscillation experiments
More informationPHY 396 K. Solutions for problem set #11. Problem 1: At the tree level, the σ ππ decay proceeds via the Feynman diagram
PHY 396 K. Solutions for problem set #. Problem : At the tree level, the σ ππ decay proceeds via the Feynman diagram π i σ / \ πj which gives im(σ π i + π j iλvδ ij. The two pions must have same flavor
More informationSection 11: Review. µ1 x < 0
Physics 14a: Quantum Mechanics I Section 11: Review Spring 015, Harvard Below are some sample problems to help study for the final. The practice final handed out is a better estimate for the actual length
More informationOscillations of neutrinos and mesons in quantum field theory
Oscillations of neutrinos and mesons in quantum field theory Mikael Beuthe Institut de Physique Théorique, Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium Abstract The controversies
More informationQuantum Light-Matter Interactions
Quantum Light-Matter Interactions QIC 895: Theory of Quantum Optics David Layden June 8, 2015 Outline Background Review Jaynes-Cummings Model Vacuum Rabi Oscillations, Collapse & Revival Spontaneous Emission
More informationFinite-size corrections to Fermi s golden rule
Hokkaido University E-mail: ishikawa@particle.sci.hokudai.ac.jp Yutaka Tobita Hokkaido University E-mail: tobita@particle.sci.hokudai.ac.jp The transition process in quantum mechanics has been studied
More informationGauge invariant quantum gravitational decoherence
Gauge invariant quantum gravitational decoherence Teodora Oniga Department of Physics, University of Aberdeen BritGrav 15, Birmingham, 21 April 2015 Outline Open quantum systems have so far been successfully
More informationSpin-Boson Model. A simple Open Quantum System. M. Miller F. Tschirsich. Quantum Mechanics on Macroscopic Scales Theory of Condensed Matter July 2012
Spin-Boson Model A simple Open Quantum System M. Miller F. Tschirsich Quantum Mechanics on Macroscopic Scales Theory of Condensed Matter July 2012 Outline 1 Bloch-Equations 2 Classical Dissipations 3 Spin-Boson
More informationHow Do Neutrinos Propagate? Wave-Packet Treatment of Neutrino Oscillation
47 Progress of Theoretical Physics, Vol. 05, No. 3, March 200 How Do Neutrinos Propagate? Wave-Packet Treatment of Neutrino Oscillation Yoshihiro Takeuchi, ) Yuichi Tazaki, S.Y.Tsai ) and Takashi Yamazaki
More information5.74 Introductory Quantum Mechanics II
MIT OpenCourseWare http://ocw.mit.edu 5.74 Introductory Quantum Mechanics II Spring 009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Andrei Tokmakoff,
More informationTwo-level systems coupled to oscillators
Two-level systems coupled to oscillators RLE Group Energy Production and Conversion Group Project Staff Peter L. Hagelstein and Irfan Chaudhary Introduction Basic physical mechanisms that are complicated
More informationSECOND PUBLIC EXAMINATION. Honour School of Physics Part C: 4 Year Course. Honour School of Physics and Philosophy Part C C4: PARTICLE PHYSICS
A047W SECOND PUBLIC EXAMINATION Honour School of Physics Part C: 4 Year Course Honour School of Physics and Philosophy Part C C4: PARTICLE PHYSICS TRINITY TERM 05 Thursday, 8 June,.30 pm 5.45 pm 5 minutes
More informationWhat We Really Know About Neutrino Speeds
What We Really Know About Neutrino Speeds Brett Altschul University of South Carolina March 22, 2013 In particle physics, the standard model has been incredibly successful. If the Higgs boson discovered
More informationDissertation. Joachim Kopp
Dissertation submitted to the Combined Faculties of the Natural Sciences and Mathematics of the Ruperto-Carola-University of Heidelberg, Germany for the degree of Doctor of Natural Sciences Put forward
More informationSearching for non-standard interactions at the future long baseline experiments
Searching for non-standard interactions at the future long baseline experiments Osamu Yasuda Tokyo Metropolitan University Dec. 18 @Miami 015 1/34 1. Introduction. New Physics in propagation 3. Sensitivity
More informationCOMMENTS UPON THE MASS OSCILLATION FORMULAS. Abstract
COMMENTS UPON THE MASS OSCILLATION FORMULAS S. De Leo (a,b), G. Ducati (b) and P. Rotelli (a) (a) Dipartimento di Fisica, INFN, Sezione di Lecce, via Arnesano, CP 193, 73100 Lecce, Italia. (b) Departamento
More informationQuantum kinetic description of neutrino oscillations arxiv: v2 [hep-ph] 27 Dec 2018
Quantum kinetic description of neutrino oscillations arxiv:1404.566v [hep-ph] 7 Dec 018 A. Kartavtsev E-mail: alexander.kartavtsev@mpp.mpg.de Abstract. Simultaneous treatment of neutrino oscillations and
More informationNon-Standard Interaction of Solar Neutrinos in Dark Matter Detectors
Non-Standard Interaction of Solar Neutrinos in Dark Matter Detectors Texas A&M University Collaboration with: J. Dent, B. Dutta, J. Newstead, L. Strigari and J. Walker Outline Motivation 1 Motivation Non-standard
More informationUnusual Interactions of Pre-and-Post-Selected Particles
EP J Web of Conferences 70, 00053 (04) DOI: 0.05 ep jconf 047000053 C Owned by the authors, published by EDP Sciences, 04 Unusual Interactions of Pre-and-Post-Selected Particles Y. Aharonov, E.Cohen, S.Ben-Moshe
More informationTime-modulation of electron-capture decay factor detected at GSI, Darmstadt
Time-modulation of electron-capture decay factor detected at GSI, Darmstadt Byung Kyu Park Department of Physics University of California, Berkeley Physics 250 March 20, 2008 Byung Kyu Park (UC Berkeley)
More informationIntroduction to particle physics Lecture 2
Introduction to particle physics Lecture 2 Frank Krauss IPPP Durham U Durham, Epiphany term 2009 Outline 1 Quantum field theory Relativistic quantum mechanics Merging special relativity and quantum mechanics
More informationSECOND PUBLIC EXAMINATION. Honour School of Physics Part C: 4 Year Course. Honour School of Physics and Philosophy Part C C4: PARTICLE PHYSICS
754 SECOND PUBLIC EXAMINATION Honour School of Physics Part C: 4 Year Course Honour School of Physics and Philosophy Part C C4: PARTICLE PHYSICS TRINITY TERM 04 Thursday, 9 June,.30 pm 5.45 pm 5 minutes
More informationFundamental of Spectroscopy for Optical Remote Sensing Xinzhao Chu I 10 3.4. Principle of Uncertainty Indeterminacy 0. Expression of Heisenberg s Principle of Uncertainty It is worth to point out that
More informationProblem 1: Spin 1 2. particles (10 points)
Problem 1: Spin 1 particles 1 points 1 Consider a system made up of spin 1/ particles. If one measures the spin of the particles, one can only measure spin up or spin down. The general spin state of a
More informationQuantum Physics III (8.06) Spring 2016 Assignment 3
Quantum Physics III (8.6) Spring 6 Assignment 3 Readings Griffiths Chapter 9 on time-dependent perturbation theory Shankar Chapter 8 Cohen-Tannoudji, Chapter XIII. Problem Set 3. Semi-classical approximation
More informationProblems for SM/Higgs (I)
Problems for SM/Higgs (I) 1 Draw all possible Feynman diagrams (at the lowest level in perturbation theory) for the processes e + e µ + µ, ν e ν e, γγ, ZZ, W + W. Likewise, draw all possible Feynman diagrams
More information6-8 February 2017 Hotel do Mar Sesimbra. Hands on Neutrinos
6-8 February 2017 Hotel do Mar Sesimbra Hands on Neutrinos Hands on Neutrinos 1 I. BRIEF HISTORY OF NEUTRINOs The neutrinowas first postulated by Wolfgang Pauli in 1930 to explain how β particles emitted
More informationPath integral in quantum mechanics based on S-6 Consider nonrelativistic quantum mechanics of one particle in one dimension with the hamiltonian:
Path integral in quantum mechanics based on S-6 Consider nonrelativistic quantum mechanics of one particle in one dimension with the hamiltonian: let s look at one piece first: P and Q obey: Probability
More informationQuantum control of dissipative systems. 1 Density operators and mixed quantum states
Quantum control of dissipative systems S. G. Schirmer and A. I. Solomon Quantum Processes Group, The Open University Milton Keynes, MK7 6AA, United Kingdom S.G.Schirmer@open.ac.uk, A.I.Solomon@open.ac.uk
More informationQuantization of the E-M field. 2.1 Lamb Shift revisited 2.1. LAMB SHIFT REVISITED. April 10, 2015 Lecture XXVIII
.. LAMB SHFT REVSTED April, 5 Lecture XXV Quantization of the E-M field. Lamb Shift revisited We discussed the shift in the energy levels of bound states due to the vacuum fluctuations of the E-M fields.
More informationSpace-Time Symmetries
Space-Time Symmetries Outline Translation and rotation Parity Charge Conjugation Positronium T violation J. Brau Physics 661, Space-Time Symmetries 1 Conservation Rules Interaction Conserved quantity strong
More informationColumbia University Department of Physics QUALIFYING EXAMINATION
Columbia University Department of Physics QUALIFYING EXAMINATION Friday, January 18, 2013 3:10PM to 5:10PM General Physics (Part II) Section 6. Two hours are permitted for the completion of this section
More informationQuantum Quenches in Extended Systems
Quantum Quenches in Extended Systems Spyros Sotiriadis 1 Pasquale Calabrese 2 John Cardy 1,3 1 Oxford University, Rudolf Peierls Centre for Theoretical Physics, Oxford, UK 2 Dipartimento di Fisica Enrico
More informationNeutrinos Lecture Introduction
Neutrinos Lecture 16 1 Introduction Neutrino physics is discussed in some detail for several reasons. In the first place, the physics is interesting and easily understood, yet it is representative of the
More informationHint of CPT Violation in Short-Baseline Electron Neutrino Disappearance
Journal of Physics: Conference Series 335 (011) 01054 doi:10.1088/174-6596/335/1/01054 Hint of CPT Violation in Short-Baseline Electron Neutrino Disappearance Carlo Giunti INFN, Sezione di Torino, and
More informationElectromagnetic modulation of monochromatic neutrino beams
Journal of Physics: Conference Series PAPER OPEN ACCESS Electromagnetic modulation of monochromatic neutrino beams To cite this article: A L Barabanov and O A Titov 2016 J. Phys.: Conf. Ser. 675 012009
More informationLepton Number Violation in Astrophysical Environments
Neutrino Mass: From the Terrestrial Laboratory to the Cosmos Amherst Center for Fundamental Interactions, December 15 2015 Lepton Number Violation in Astrophysical Environments Vincenzo Cirigliano Los
More informationChapter 29. Quantum Chaos
Chapter 29 Quantum Chaos What happens to a Hamiltonian system that for classical mechanics is chaotic when we include a nonzero h? There is no problem in principle to answering this question: given a classical
More informationInterference Between Distinguishable States. Thomas Alexander Meyer
Interference Between Distinguishable States Thomas Alexander Meyer Interference effects are known to have a dependence upon indistinguishability of path. For this reason, it is accepted that different
More informationDecays, resonances and scattering
Structure of matter and energy scales Subatomic physics deals with objects of the size of the atomic nucleus and smaller. We cannot see subatomic particles directly, but we may obtain knowledge of their
More informationSterile neutrinos. Stéphane Lavignac (IPhT Saclay)
Sterile neutrinos Stéphane Lavignac (IPhT Saclay) introduction active-sterile mixing and oscillations cosmological constraints experimental situation and fits implications for beta and double beta decays
More informationThe Exchange Model. Lecture 2. Quantum Particles Experimental Signatures The Exchange Model Feynman Diagrams. Eram Rizvi
The Exchange Model Lecture 2 Quantum Particles Experimental Signatures The Exchange Model Feynman Diagrams Eram Rizvi Royal Institution - London 14 th February 2012 Outline A Century of Particle Scattering
More informationCollapse and the Tritium Endpoint Pileup
Collapse and the Tritium Endpoint Pileup arxiv:nucl-th/9901092v2 4 Feb 1999 Richard Shurtleff January 23, 1999 Abstract The beta-decay of a tritium nucleus produces an entangled quantum system, a beta
More informationNeutrino Physics After the Revolution. Boris Kayser PASI 2006 October 26, 2006
Neutrino Physics After the Revolution Boris Kayser PASI 2006 October 26, 2006 1 What We Have Learned 2 The (Mass) 2 Spectrum ν 3 ν 2 ν 1 } Δm 2 sol (Mass) 2 Δm 2 atm or Δm 2 atm ν ν 2 } Δm 2 sol 1 ν 3
More information1 Neutrinos. 1.1 Introduction
1 Neutrinos 1.1 Introduction It was a desperate attempt to rescue energy and angular momentum conservation in beta decay when Wolfgang Pauli postulated the existence of a new elusive particle, the neutrino.
More informationconventions and notation
Ph95a lecture notes, //0 The Bloch Equations A quick review of spin- conventions and notation The quantum state of a spin- particle is represented by a vector in a two-dimensional complex Hilbert space
More informationECE 487 Lecture 6 : Time-Dependent Quantum Mechanics I Class Outline:
ECE 487 Lecture 6 : Time-Dependent Quantum Mechanics I Class Outline: Time-Dependent Schrödinger Equation Solutions to thetime-dependent Schrödinger Equation Expansion of Energy Eigenstates Things you
More informationOn the detection of kev scale neutrino dark matter in β decay
On the detection of kev scale neutrino dark matter in β decay experiment Wei LIAO U East China University of Science and Technology uànóœæ Workshop CIAS Meudon 2012 6 June 2012, Paris Content Content:
More informationLecture 4: Entropy. Chapter I. Basic Principles of Stat Mechanics. A.G. Petukhov, PHYS 743. September 7, 2017
Lecture 4: Entropy Chapter I. Basic Principles of Stat Mechanics A.G. Petukhov, PHYS 743 September 7, 2017 Chapter I. Basic Principles of Stat Mechanics A.G. Petukhov, Lecture PHYS4: 743 Entropy September
More informationQuantum Confinement in Graphene
Quantum Confinement in Graphene from quasi-localization to chaotic billards MMM dominikus kölbl 13.10.08 1 / 27 Outline some facts about graphene quasibound states in graphene numerical calculation of
More informationQuantum Physics II (8.05) Fall 2002 Outline
Quantum Physics II (8.05) Fall 2002 Outline 1. General structure of quantum mechanics. 8.04 was based primarily on wave mechanics. We review that foundation with the intent to build a more formal basis
More informationA Minimal Uncertainty Product for One Dimensional Semiclassical Wave Packets
A Minimal Uncertainty Product for One Dimensional Semiclassical Wave Packets George A. Hagedorn Happy 60 th birthday, Mr. Fritz! Abstract. Although real, normalized Gaussian wave packets minimize the product
More informationFlavor oscillations of solar neutrinos
Chapter 11 Flavor oscillations of solar neutrinos In the preceding chapter we discussed the internal structure of the Sun and suggested that neutrinos emitted by thermonuclear processes in the central
More informationAn Inverse Mass Expansion for Entanglement Entropy. Free Massive Scalar Field Theory
in Free Massive Scalar Field Theory NCSR Demokritos National Technical University of Athens based on arxiv:1711.02618 [hep-th] in collaboration with Dimitris Katsinis March 28 2018 Entanglement and Entanglement
More informationThe Daya Bay and T2K results on sin 2 2θ 13 and Non-Standard Neutrino Interactions (NSI)
The Daya Bay and TK results on sin θ 13 and Non-Standard Neutrino Interactions (NSI) Ivan Girardi SISSA / INFN, Trieste, Italy Based on: I. G., D. Meloni and S.T. Petcov Nucl. Phys. B 886, 31 (014) 8th
More informationQuantum mechanics in one hour
Chapter 2 Quantum mechanics in one hour 2.1 Introduction The purpose of this chapter is to refresh your knowledge of quantum mechanics and to establish notation. Depending on your background you might
More informationBeyond Standard Model Effects in Flavour Physics: p.1
Beyond Standard Model Effects in Flavour Physics: Alakabha Datta University of Mississippi Feb 13, 2006 Beyond Standard Model Effects in Flavour Physics: p.1 OUTLINE Standard Model (SM) and its Problems.
More informationPerturbation Theory. Andreas Wacker Mathematical Physics Lund University
Perturbation Theory Andreas Wacker Mathematical Physics Lund University General starting point Hamiltonian ^H (t) has typically noanalytic solution of Ψ(t) Decompose Ĥ (t )=Ĥ 0 + V (t) known eigenstates
More informationQuantum Measurements: some technical background
Quantum Measurements: some technical background [From the projection postulate to density matrices & (introduction to) von Neumann measurements] (AKA: the boring lecture) First: One more example I wanted
More information1 The muon decay in the Fermi theory
Quantum Field Theory-I Problem Set n. 9 UZH and ETH, HS-015 Prof. G. Isidori Assistants: K. Ferreira, A. Greljo, D. Marzocca, A. Pattori, M. Soni Due: 03-1-015 http://www.physik.uzh.ch/lectures/qft/index1.html
More information1 Equal-time and Time-ordered Green Functions
1 Equal-time and Time-ordered Green Functions Predictions for observables in quantum field theories are made by computing expectation values of products of field operators, which are called Green functions
More informationPHYS 508 (2015-1) Final Exam January 27, Wednesday.
PHYS 508 (2015-1) Final Exam January 27, Wednesday. Q1. Scattering with identical particles The quantum statistics have some interesting consequences for the scattering of identical particles. This is
More information