centenary of giulio racah jerusalem 2010 guidance by …in 1950 all particle physicists knew that...

Centenary of Giulio Racah

Jerusalem 2010

Guidance by Racah on the path from Experiment to Theory

As a graduate student in experimental physics at Princeton

I heard lectures on group theory from all the experts

Wigner, Weyl and Artin showed us fancy mathematics; Nobody showed how group theory could be useful

Artin insisted group theory was interesting mathematics and should never be used

Racah showed me that group theory was useful and how to use it

Racah started with experimental data

He was an amazing calculator and looked for systematics in the data

At our weekly Thursday joint seminar at the Weizmann Institute

Racah would come to Rehovot bringing his students

He would immediately disappear to the computer until called for the beginning of the seminar

I remember a series of seminars that he gave that went on for several weeks

Finally as he looked at the blackboard completely covered with equations he said

Aha! This is a recursion formula for an XXX function of the YYYth kind.

Racah found general results by brute force calculations manipulating experimental data

Only then would he put it all together in an elegant theory

Unfortunately Racah’s papers were too complicated for experimentalists like me to understand.

1

Racah’s physical insight, clear in his lectures and discussions, was lost in his published papers.

My first two books were “Racah for Pedestrians”

They put down what experimentalists needed in a way they could understand

Beta decay for Pedestrians

In 1956 the discovery of parity violation set off a series of new experiments

But all available literature on angular correlations assumed parity was conserved

There was no guidance in the literature for how experimenters should design and interpret parity experiments

Lectures I gave explaining Racah to experimenters turned into this book

Lie Groups for Pedestrians

In 1950 all particle physicists knew that group theory was useless for high energy physics

Great particle theorists were at the Princeton Institute for Advanced Study

They did not come to hear Racah’s later famous lectures on Group Theory and Spectroscopy

Shelly Glashow was one crucial exception

Glashow’s paper with Rosenfeld showed how the SU(3) group theory could make experimental predictions

His paper with Sakurai predicted the existence of a particle now called the Ω−

There was an immediate demand for me to give lectures explaining what I had learned from Racah.

These ended up in a book which taught many particle physicists what they needed to know

2

Applying what I learned from Racah to neutrino oscillations

Look at experiments!

Stay away from fancy theory. This can come only after understanding the experiment.

Neutrino oscillation textbooks and theoretical papers ignore Racah’s approach

They develop fancy theories without understanding the experiment.

Text books tell us ν at rest with definite flavor is a coherent mixture of energy eigenstates.

Interference between these states produces oscillations in time between different flavors.

The ν’s oscillate in time as coherent mixtures of states with different energies.

Text books don’t tell us that no experiment has ever seen a ν at rest

No experiment will ever see an oscillating neutrino at rest.

You cannot Lorentz transform an observed oscillating neutrino to its rest frame.

There is no rest frame. Each component with a different mass has a different rest frame.

Textbooks don’t tell how such an oscillation is created or observed in any real experiment.

Text books don’t tell us that no experiment has ever measured a time of oscillation.

3

Following Racah’s approach to Darmstadt GSI oscillations

Experiment observes radioactive ion circulating in storage ring

Oscillations in decay rate give information about ν masses without detecting the ν

Theoretical papers saying that this is impossible are all wrong!

1. They do not follow Racah’s approach

2. They do not understand what is really measured in this experiment

3. They describe gedanken experiments which have no relation to the real experiment

4. They do not examine the right physics of the experiment

5. They give the right answers to the wrong questions

What are the Right Questions?

What is the Right Physics? What is actually measured?

Need only elementary quantum mechanics and time-dependent perturbation theory

High brow quantum field theory useless

4

What is the Right Physics? What is actually measured?

1. Ion monitored at regular intervals during passage around storage ring.

2. Each monitoring collapses wave function (or destroys entanglement phase).

3. Decay of initial state measured by disappearance between successive monitoring.

4. Probability of decay in interval between two collapses is measured

5. Decay probability measured in interval between two points in space-time

6. Two points are at the same point in space in laboratory system

7. Much more complicated in other systems. Need to work in the lab frame

8. Relativistic quantum field theory useless here

What are the Right Questions?

1. What determines measured decay probability

2. Decay probability in such short time given by Fermi “Golden Rule”

3. Departure from exponential dependence indicates change of initial state

4. Depends upon propagation of initial state in circulation around storage ring.

(a) Must consider relativistic dynamics of motion in storage ring

(b) Experiment monitors initial state at same point and different times in laboratory

(c) In all other frames these are different space points and different times

(d) Initial state wave function accelerated between relativistic opposite velocities

(e) Rest frame of ion oscillates and undergoes acceleration - need general relativity?

5

Our approach here starts from the end

No oscillations if we know ν mass; Why can’t we know it?

Why are ν oscillation experiments not “missing mass” experiments?

No oscillations if conservation laws determine mass of unobserved neutrino

Detector destroys all interference between states with different energies

Final state must be mixtures of states with same energy and different momentum

We start with a well-defined initial state of a radioactive ion

Why can’t conservation laws for the decay to final state determine neutrino mass?

Relativistic time dilatation as in the famous twin paradox

Components of an ion wave packet having slightly different momenta in laboratory

Have slightly different lifetimes in the laboratory system.

Radioactive ion moving with relativistic velocity in laboratory system has longer lifetime

Than one at rest when both lifetimes measured in laboratory system.

Components of an ion wave packet having slightly different momenta in laboratory

Have slightly different lifetimes in the laboratory system.

Amplitudes for decays of these different components will be different

Will have different relative phases as a function of time in the laboratory.

One of the twins is “younger than the other”

This phase difference can produce oscillations in time behavior of decay probability.

6

Why GSI experiment can be a major breakthrough in ν oscillations

Answers two principal difficulties of neutrino experiments

1. Ordinary neutrino oscillation experiments are difficult because

(a) The neutrino absorption cross section is small.

The number of neutrino events actually used in ordinary experiments is many orders of magnitude

smaller than the number events creating the neutrinos.

(b) The oscillation wave lengths are so large that it is difficult to actually follow one oscillation

period in any experiment.

2. This experiment opens up a new line for dealing with these difficulties

(a) The oscillation is measured without detecting the neutrino.

Detection of every neutrino creation event avoids the losses from the low neutrino absorption cross

section.

(b) The long wave length problem is solved by having the radioactive source move a long distance

circulating around in a storage ring.

7

How can a weak decay not be exponential?

Non-exponential time-dependence observed in decay probability

Although counterintuitive, it follows naturally from being a “watched pot” experiment.

Initial state of ion monitored during passage around a storage ring,

Affirms that ion has not yet decayed

Like“Schroedinger cat” experiment with door opened at regular intervals

Repeated measurement of whether the cat is still alive.

Initial state - free ion moving with repeated detection in fields of apparatus

Time interval t between entry into apparatus and last observation before decay

Time t′ between last monitoring and decay negligible

Initial state |i(t)〉 - Time dependent wave packet with components of different energies.

Relative phases determined by localization in space at point of entry into apparatus.

Hamiltonian Ho describes the motion of a free initial ion

In electromagnetic fields constraining its orbit in storage ring.

Evolution of initial state and transition matrix element for decay to final state |f〉|i(t)〉 = eiHot |i(0)〉 ; 〈f |T |i(t)〉 = 〈f |TeiHot |i(0)〉Transition probability per unit time at time t given by Fermi’s Golden Rule,

W (t) =2π

h| 〈f |T |i(t)〉 |2ρ(Ef ) =

2π

h| 〈f |TeiHot |i(0)〉 |2ρ(Ef )

8

Probability Pi that ion is still in initial state and not decayed satisfies differential equation

d

dtPi = −W (t)Pi;

d

dtlog(Pi) = −W (t); Pi = e−

∫W (t)dt

Exponential decay only for time-independent transition matrix element

Time dependence of decay not necessarily exponential.

Transition probability depends upon propagation of initial state during time t between the entry of the

ion into apparatus and time of the decay.

Time-dependent perturbation theory suggests presence of decay amplitude before the decay

Continued observation of initial ion before decay collapses wave function

Rules out any influence of any final state amplitude on decay process.

Time dependence depends only on propagation of initial state independent of final state created only

at the decay point.

No violation of causality; no information about final state before decay.

9

Variation in time of the transition probability for Darmtadt Experiment

Simplified model for propagation and oscillating behavior of wave function in time

Initial “Mother” ion wave packet with two components |i1(t)〉 and |i2(t)〉Slightly different unperturbed energies E and E + δE

Both decay into same final state. Time development before the decay

|i1(t)〉+ |i2(t)〉 = eiHot[|i1(0)〉+ |i2(0)〉] = eiEt[|i1(0)〉+ eiδEt |i2(0)〉]

Time t = 0 time of entry into the apparatus

Ho is unperturbed Hamiltonian describing motion of wave packet in electromagnetic fields in storage ring.

The transition from the initial state to a final state denoted by |f〉 is

W (t) =2π

h| 〈f |TeiEt[|i1(0)〉+ eiδEt |i2(0)〉]|2ρ(Ef )

h

2π·W (t) =| 〈f |T |i1(0)〉 |2 + | 〈f |T |i2(0)〉 |2 +2 cos(δEt + δφ) | 〈f |T |⊂2 (0)〉 · 〈⊂1 (0)|T |f〉 |

where δφ denotes the relative phase of the two transition matrix elements.

Decay probability oscillates in time with a frequency δE

Observable as a sinusoidal modulation of an exponential decay and explains Darmstadt oscillations.

10

Evaluation of oscillation frequency δE

Depends upon energy levels of unperturbed hamiltonian H0 including fields creating orbit in storage ring

A very crude approximation obtains a result with correct order of magnitude.

Consider the transition for an initial state having momentum ~P and energy E.

Final state has recoil ion with momentum ~PR and energy ER and ν with energy Eν and momentum ~pν .

ER = E − Eν ; ~PR = ~P − ~pν ; M2 + m2 −M2R = 2EEν − 2~P · ~pν

where M , MR and m denote respectively the masses of the mother and daughter ions and the neutrino.

Simplified initial state for “mother” ion with two components∣∣∣~P

⟩and

∣∣∣~P + δ ~P⟩

having momenta ~P and

~P + δ ~P with energies E and E + δE.

|i1(t)〉+ |i2(t)〉 =∣∣∣~P , E

⟩+ eiδEt ·

∣∣∣(~P + δ ~P ), (E + δE)⟩

The final state |f〉 has two components with neutrino momenta pν and pν + δpν with energies Eν and

Eν + δEν together with a recoil ion having the same momentum and energy for both components.

A very crude approximation Relates δE to the momentum difference δP .

It assumes momentum conservation in the transition, uses the relation between momentum and energy of

a free particle, neglects transverse momenta and the effects of evolution of the wave packet during passage

around the storage ring through straight sections, bending sections and focusing electric and magnetic fields.

11

The changes produced by a small change ∆(m2) in the squared neutrino mass satisfy the relation

∆(m2)2EδE

≈ −1; δE ≈ −∆(m2)2γM

γ is the Lorentz factor (E/M),

We have used approximations and noted that momentum conservation in the transition requires δpν = δP .

δEν = 0 since the two final neutrino components must have the same energy to enable coherence.

Then δE 6= δEν = 0 and we are violating energy conservation.

The period of oscillation δt is then

δt ≈ 2π

δE=

4πγM

∆(m2); ∆(m2) =

4πγM

δt≈ 2.75∆(m2)exp

12

Essential features of the “watched pot” experiment

Transition restricted to tiny region in space-time

1. The interval in space-time (δx, δt) between two points in space-time

(a) Where the initial state was last observed not to have decayed

(b) where the decay was observed

known to be small with sufficient precision to be approximated by a point.

2. The transition matrix element 〈f, t|T |i, t〉 is between the values of the initial and final state wave

functions at this point.

3. The final state wave function includes pairs of neutrino mass eigenstates with the same energy but

different masses and momenta.

4. If both energy and momentum of the final state neutrino are observable there is no coherence and no

oscillations.

5. Energy of final neutrino state observable; neutrino momentum not observable

6. The transitions to final neutrino states which have the same energy and different momenta are coher-

ent. The final state wave function includes pairs of such states with well defined momentum differences

and relative amplitudes and relative phases.

7. We assume that momentum is conserved in the transition. The momentum difference between com-

ponents of the initial state that can interfere coherently is equal to the momentum difference between

the neutrino mass eigenstates.

13

8. The transition matrix element therefore is determined by the overlap of pairs of components in the

initial and final state wave functions with a well defined momentum difference; i.e. by their amplitudes

and relative phases.

9. The relative amplitudes and phases of these pairs of components in the final state wave function at

the decay point known from the parameters of the neutrino mass matrix and the requirement that

the final state is an electron neutrino. These are independent of the point in space-time where the

decay occurs.

10. The relative amplitudes and phases of these pairs of components in the initial state wave function

known at the entry into the apparatus. The changes in these values between the entry and decay

points determined by propagation through magnetic fields in storage ring over the space-time interval

between these points

11. Since the pairs of initial wave functions with different momenta have different energies, a phase

difference between these components occurs which depends linearly on the energy difference. The

propagation of these waves around the storage ring can introduce additional phase differences which

remain to be calculated using the precise parameters describing the motion through the storage ring.

14

Why are neutrino oscillations observable?

Text books don’t tell you

Textbooks tell us that a coherent linear combination of ν masses will oscillate in time

Nobody measures time and textbooks don’t tell us why and how such oscillations are observable

You won’t find the answer to this in relativistic quantum field theory

You have to look at the experiments that detect the neutrino

Three problems missed in the textbooks

1. What is the connection between the time in the textbook and the distance measured in the experiment

(a) The distance is the product of the time and the velocity

(b) But neutrinos with different masses travel with different velocities between source and detector.

(c) Arguments about how to handle this time difference continually arise and swept under the rug.

2. In a “missing mass experiment” the mass of an unobserved ν is determined by conservation laws.

(a) A neutrino is detected by charge exchange with a nucleon and the emission of a lepton

(b) The ν energy is equal to the energy change in the detector nucleon plus the lepton energy

(c) The ν momentum is equal to the momentum change in the nucleon plus the lepton momentum

(d) If both energy and momentum changes in the detector are observable; the ν mass is observable.

(e) If the ν mass is observable there can be no oscillations between states with different masses

(f) ν oscillations are observed; Something must be unobservable.

3. Are oscillations produced between neutrinos with different energies or with different momenta?

15

Answers to the problems

1. Focus on what is directly measured in real experiments, not gedanken experiments

(a) What is measured is the distance between some reference point and the detector.

(b) The neutrino is a wave packet passing the detector

(c) The probability density of the wave passing the detector gives the probability of detection

(d) Estimates of time are unreliable. Group velocity tells only position of center of wave packet.

(e) The exact time of the detection has statistical fluctuations.

2. What prevents us from knowing the energy and momentum of the neutrino?

(a) Mossbauer physics and Dicke Superradiance needed to understand neutrino oscillations

(b) The detector is a massive object with effectively infinite mass fixed in space in laboratory system

(c) Detector absorbs missing momentum with zero energy transfer like crystal in Mossbauer Effect

(d) The ν momentum is NOT equal to momentum change in detector nucleon plus lepton momentum

(e) The ν energy IS equal to energy change in detector nucleon plus lepton energy

(f) The detector is a quantum system with definite energy levels in the laboratory

3. Absorption of two ν states with different momenta and same energy produce the same detector

transition

(a) No measurement on final state can determine the momentum of the incident neutrino

(b) Neutrino states with the same energy are absorbed coherently by a detector

(c) Neutrino states with different energy are all incoherent.

16

Neutrinos are generally detected by a charge exchange scattering with a nucleon in a detector.

ν + p → µ+ + n; ν + n → µ− + p

1. Energy and momentum conservation in lepton nucleon system gives“missing mass” experiment.

2. Conservation laws determines the neutrino mass and there are no oscillations

3. If the mass of the neutrino is observable, there are no oscillations

A coherent linear combination of ν masses enters a detector

1. This is a two-slit experiment in neutrino energy-momentum space.

2. If we know the energy and momentum of the neutrino, we know its path in energy-momentum space

3. If we know the energy and momentum of the neutrino, we know its mass - No interference

4. ν oscillations are observed; Something must be unobservable.

What prevents us from knowing the energy and momentum of the neutrino?

You won’t find the answer in relativistic quantum field theory

Text books don’t tell us what is unobservable

Condensed matter physics, Mossbauer phywics and Dicke superradiance needed to understand ν oscillations

1. A neutrino enters a detector, changes the charge of a nucleon and emits a charged lepton

2. If energy and momentum are conserved the ν mass is observable; there can be no oscillations

(a) The ν energy is equal to the energy change in the detector nucleon plus the lepton energy

(b) The ν momentum is equal to the momentum change in the nucleon plus the lepton momentum

(c) If both the energy and momentum changes in the detector are observable; No oscillations.

17

Mossbauer physics and Dicke Superradiance needed to understand neutrino oscillations

The detector is a massive object with effectively infinite mass fixed in space in laboratory system

Detector absorbs missing momentum with zero energy transfer like crystal in Mossbauer Effect

The ν momentum is NOT equal to momentum change in detector nucleon plus lepton momentum

The ν energy IS equal to energy change in detector nucleon plus lepton energy

The detector is a quantum system with definite energy levels in the laboratory

Absorption of two ν states with different momenta and same energy produce the same detector transition

No measurement on final state can determine the momentum of the incident neutrino

Dicke superradiance arises when several initial states can produce the same final state

ν detection is a “which-path” or “two-slit” experiment in momentum space

Why should we care about this?

We know the right answer to how to interpret ν oscillation results.

1. We want our students to understand the basic physics

2. We want to go beyond the accepted experments and know how to look for new experiments

3. Can ν masses be measured without etecting the ν in a missin mass experiment?

4. Are Darmstadt oscillations real? Can they teach us something new?

18

Why do neutrinos oscillate?

Text books don’t tell you

If the mass of the neutrino is observable, there are no oscillations

If we know the energy and momentum of the neutrino, we know its mass - No oscillations

What prevents us from knowing the energy and momentum of the neutrino?

1. A neutrino enters a detector, changes the charge of a nucleon and emits a charged lepton

2. If energy and momentum are conserved the ν mass is observable; there can be no oscillations

(a) The ν energy is equal to the energy change in the detector nucleon plus the lepton energy

(b) The ν momentum is equal to the momentum change in the nucleon plus the lepton momentum

(c) If both the energy and momentum changes in the detector are observable; No oscillations.

3. ν oscillations are observed; Something must be unobservable.

Text books don’t tell us what is unobservable

Mossbauer physics and Dicke Superradiance needed to understand neutrino oscillations

The detector is a massive object with effectively infinite mass fixed in space in laboratory system

The ν momentum is NOT equal to momentum change in detector nucleon plus lepton momentum

Detector absorbs missing momentum - like crystal in the Mossbauer Effect

Absorption of two ν states with different momenta produce the same detector transition



19

Neutrinos emitted in a π − µ decay are linear combinations of neutrino mass eigenstates with different

masses, different energies and different momenta.

When these neutrinos were first detected at BNL, only muons were observed, no electrons

Experimentally observed absence of electrons explainable only if electron amplitudes

Received at detector from different ν mass eigenstates are coherent and exactly cancel.

How do relative phases of relevant amplitudes cancel the production of an electron?



The ν’s oscillate as coherent mixtures of states with different energies.


Don’t tell how such an oscillation is created or observed in any real experiment.

Sufficient information not available to determine ν mass from energy and momentum conservation.

Coherence or interference between different ν mass eigenstates not observable in “missing mass” experiment

with unobserved ν mass uniquely determined by other measurements and conservation laws.

A missing mass experiment was not performed. Detector nucleon wave function must cancel probability

of finding nucleon outside detector at all times

Interference between components with different momenta provides cancelation at each energy

Components of incident ν with same energy and different momenta absorbed by detector nucleon eigenstate

Leavs no trace identifying momentum of incident ν.

20

Why do neutrinos oscillate? Why are neutrino oscillations observable?

Relativistic quantum field theory is useless

1. It tells the wave function emitted from a weak decay

2. It does not tell what is observable in a given experiment

Mossbauer effect physics and Dicke superradiance needed to describe ν oscillations

Neutrinos are detected by charge exchange with a nucleon producing a charged lepton

1. The nucleon is bound in a massive detector with essentially infinite mass

(a) The detector can absorb finite momentum with zero energy transfer

(b) Energy is conserved in the absorption of the neutrino by a nucleon

(c) Momentum can be carried away by the whole detector, like the crystal in the Mossbauer effect

2. Transition to any final nucleon-lepton state leaves no trace of the entering neutrino momentum

3. Neutrino detection is a “two slit” or ”which-path” experiment in momentum space

Dicke superradiance in the neutrino detection.

1. Components of entering neutrino with same energy and different momenta produce same final state.

2. Transition amplitude depends on relative phases

3. State with maximum constructive constructive interference is “Dicke superradiant” state

4. States orthogonal to superradiant state are called subradiant states,

Relative phases between states with different ν momenta change with distance between source and detector

Phase changes between superradiant and subradiant states produce observed ν oscillations with distance

21

Symmetry in Physics and More

Centenary of Giulio Racah

Guidance by Racah on the path from Experiment to Theory

Racah showed me that group theory was useful and how to use it

Racah was an amazing calculator, started with experimental data and looked for systematics

Racah found general results by brute force calculations manipulating experimental data

Only then would he put it all together in an elegant theory

My first two books were “Racah for Pedestrians”

They put down what experimentalists needed in a way they could understand

Beta decay for Pedestrians; Lie Groups for Pedestrians

What I learned from Racah for ν oscillations; Look at experiments!

Stay away from fancy theory. This can come only after understanding the experiment.

Neutrino oscillation textbooks and theoretical papers ignore Racah’s approach

They develop fancy gedanken theories for ν at rest with definite flavor oscillating in time


You cannot Lorentz transform an observed oscillating neutrino to its rest frame.

There is no rest frame. Each component with a different mass has a different rest frame.

Relativistic quantum field theory useless for neutrino oscillations

Need condensed matter physics, Mossbauer Effect and Dicke Superradiance

22

Follow Racah. See what we learn from experiment

The original Lederman-Schwartz-Steinberger neutrino experiment found only muons

1. Neutrinos emitted in π − µ decay produced no electron neutrinos

2. At least two neutrino mass eigenstates are emitted in π − µ decay

3. At least one of them can produce an electron in a neutrino detector.

Neutrino from π → µν enters detector, changes charge of nucleon and emits muon and no electron.

Why are no electrons observed? Why are neutrino oscillations observable?

A ν mass eigenstate can produce either an electron or a muon. The electron amplitude must be canceled

By interference between amplitudes produced by an entering coherent mixture of ν mass eigenstates.

If energy and momentum are conserved there is no coherence and no way to cancel the emitted electron.

1. Entering ν’s with different masses have different four-momenta

2. They produce different final states with different four-momenta

3. If four-momentum is conserved the ν mass is observable; there can be no oscillations

23

To answer this paradox. Follow Racah and look at the experiment

The neutrino is detected by interaction with a nucleon in a detector

What does experiment tell us about the wave function of the detector nucleon?

The probability of finding the nucleon outside the detector must be zero!

This constraint is crucial to understanding interference between neutrinos with different masses

You won’t find it in any text books or fancy field theories

To understand coherence of two incident neutrinos with a momentum difference δp

We find a Debye-Waller factor e−δp2·L2 ≈ 1 if δp2 · L2 ¿ 1

Four-momentum is not conserved because the nucleon is bound in a detector

Detector has effectively infinite mass, is confined to a definite region in space for all times

1. Nucleon can absorb momentum without energy transfer like the crystal in the Mossbauer effect.

2. Neutrinos with same energy and a momentum difference are absorbed coherently by detector.

3. Two states with different four-momenta but the same energy can produce the same final state

(a) Transition amplitude for combination of these two states depends upon relative phase

(b) Combination with maximum constructive interference is called “Dicke superradiant”

“Missing momentum” absorbed without energy transfer by whole detector

Like photon momentum by whole crystal in the Mossbauer effect.

Missing momentum prevents determination of ν mass by conservation laws.

Mossbauer recoilless transition and Dicke superradiance crucial for understanding ν coherence

24

Nucleon makes a transition |A〉 → |B〉Between initial detector nucleon eigenstate |A〉 to final state |B〉 with momentum transfer ~p

Transition matrix element proportional to the momentum transfer matrix element,

〈B|T |A〉 ∝ 〈B| ei~p· ~X |A〉~X denotes the distance between the position of the nucleon and center of detector.

For a small change δ~p the leading term in the expansion of the small parameter δp

〈B| ei~p· ~X |A〉 − 〈B| ei[~p+δ~p]· ~X |A〉〈B| ei~p· ~X |A〉

≈ 12· 〈B| [δ~p

2 · ~X2] · ei~p· ~X |A〉〈B| ei[~p· ~X] |A〉

≤ δp2 · L2

2

L denotes the length of the detector.

If the size of the detector is much smaller than the oscillation wave length δp2 · L2 ¿ 1

Change by amount δp in ν momentum will not be detected by measuring the transition |A〉 → |B〉If absorption of two ν states with slightly different momenta can produce same change |A〉 → |B〉

There is coherence. One only sees that there was a transition from A to B.

One cannot know which neutrino mass produced the transition

The momentum difference is taken up by the whole detector.

Neutrino detection is “two-slit” or “which-path” experiment in momentum space

Contributions to final state amplitude via different paths are coherent and interference produces oscillations.

Oscillations are produced by interference between ν states with same energy and different momenta.

25

Asymmetry between energy and momentum

1. Crucial to understanding ν oscillation experiments

2. Arises from asymmetry between space and time in detector nucleon wave function

3. Probability for finding nucleon outside detector spatial region vanishes for all times

4. Detector nucleon wave function vanishes in space only outside detector but for all times.

5. Crucial constraint has no covariant description and valid only in laboratory frame

6. Relativistic quantum field theory is useless; Covariance misses asymmetry

ν’s with different momenta travel from source to detector, interfere and produce oscillations

1. Coherent absoption of ν’s with same energy, different masses and momenta

2. Produce same transition in detector eigenstates with same energy that vanish outside detector

3. No trace of ν momentum implies coherence between different momenta

4. A ν created in the decay of a one-particle state cannot oscillate if energy and momentum are conserved.

Detectors in experiments observing ν oscillations do not measure time

Destroy all interference between states with different energies

1. The detector nucleon is confined to a region smaller than the oscillation wave length

2. The nucleon wave function must vanish outside the detector for all times

3. Interference needed between different momentum eigenstates with same energy

4. Interference between different energy amplitudes would oscillate outside detector

26

Basic quantum mechanics of what is observable in neutrino detection

ν observed in experiments by interactions with nucleons in detector at rest in laboratory system.

1. The probability of finding the detector nucleon outside the detector must vanish for all times.

2. The state of the detector nucleon in quantum mechanics is described by a wave function or den-

sity matrix which gives a time-independent vanishing probability for finding the nucleon outside the

detector.

3. The density matrix describing the detector nucleon must have coherence and interference between

components with different momenta at each energy which cancel out the probability of finding the

nucleon outside the detector.

4. Components of an incident neutrino with the same energy and different momenta can produce coherent

transition amplitudes between two detector nucleon states that both have a vanishing probability of

finding the nucleon outside the detector.

5. The momentum of the neutrino that produced the transition in the detector is not observable.

6. Neutrino detection is a “two-slit” or “which-path” experiment in momentum space.

These properties of the detector nucleon in the laboratory system

1. Crucial for the description of neutrino oscillations

2. The position of the detector must be known with error smaller than oscillation wave length

3. The momentum uncertainty is large enough produce oscillations

Not simply described by relativistic quantum field theory

27

Momentum conservation is violated in the ν absorption by a nucleon in the detector

Detector absorbs missing momentum - like crystal in the Mossbauer Effect

Absorption of two ν states with different momenta produce the same detector transition



1. Neutrinos with two different momenta can produce the same final state in a detector

2. No measurement can determine which neutrino momentum produced the transition

3. States with different momenta are coherent

Neutrino oscillations arise from interference between states having different momenta

Two different states with different momenta in an initial state produce same final state in a transition

1. The transition probability depends upon the relative phase of the two initial states

2. A Dicke superradiant state can be defined in which the relative phase is chosen to maximize the

constructive interference and maximally enhance the transition probability.

3. States orthogonal to the superradiant state are called subradiant states. They have destructive

interference between their components with different momenta and suppressed transition probability.

4. The initial state is a linear combination of superradiant and subradiant states.

5. Since the two initial states have different momenta their relative phase varies with distance and

changes their linear combination between superradiant and subradiant values.

No oscillations observable from interference between states having different energies

28

The original Lederman-Schwartz-Steinberger neutrino experiment found only muons

1. Neutrinos emitted in π − µ decay produced no electron

2. At least two neutrino mass eigenstates are emitted in π − µ decay

3. At least one of them can produce an electron in a neutrino detector.

Experimentally observed absence of electrons explainable only if electron amplitudes

Received at detector from different ν mass eigenstates are coherent and exactly cancel.

Neutrinos emitted in a π − µ decay are linear combinations of neutrino mass eigenstates with different

masses, different energies and different momenta.

How do relative phases of relevant amplitudes cancel the production of an electron?

This can only be understood by investigating the quantum mechanics of the detector.



The ν’s oscillate as coherent mixtures of states with different energies.


Don’t tell how such an oscillation is created or observed in any real experiment.

Sufficient information not available to determine ν mass from energy and momentum conservation.

Coherence or interference between different ν mass eigenstates not observable in “missing mass” experiment

with unobserved ν mass uniquely determined by other measurements and conservation laws.

A missing mass experiment was not performed.

29

Why different treatments of energy and momentum give the same answer

Consider a simplified two-component ν state with two components

Momenta ~P and ~P + δ ~P with energies E and E + δE and squared masses m2 and m2 + ∆(m2).

Changes produced by a small change ∆(m2) in the squared neutrino mass satisfy the relation

E2 = P 2 + m2; 2EδE ≈ 2PδP + ∆(m2)

Neutrinos travel distance X in time t; phases φ(E, P ) and φ(E + δE, P + δP ) satisfy

φ(E,P ) = P ·X − E · t; φ(E + δE, P + δP ) = (P + δP ) ·X − (E + δE) · tIn realistic experiments measurements are made a definite distance X from the source

Time t of detection of ν not measured; estimated using group velocity PE of neutrino wave packet

Transit time tw of the center of the wave packet traversing distance X is tW = X · EP .

The relative phase δφ between the two components is given by

δφ(X) = δP ·X − δE · t ≈ X · [δP − δE · EP

]− δE · (t− tW ) ≈ − X2P ·∆(m2)− 2δE · (t− tW )

If δE = 0 or we chose t ≈ tW

Relative phase δφ(X) depends only on squared mass difference ∆(m2) between mass eigenstates

Independent of particular chosen values of δP and δE as long as these satisfy 2EδE ≈ 2PδP + ∆(m2)

Choosing states with the same energy and different momenta

Gives the same result as choosing states with the same momentum and different energies.

Quantum fluctuations t− tW in the transit time to the detection point do not affect this result

30

Following Racah: What have we learned from oscilation experiments?

1. ν oscillations observed in experiments as a function of distance between source and detector.

2. The neutrino emitted from the decay is a coherent mixture of states with different four-momenta.

3. The neutrinos are detected in a large many-body system with finite spatial dimensions and effecively

infinite mass in thermal equilibrium with an environment at rest in the laboratory system.

4. The detector is a quantum-mechanical system with definite energy levels.

(a) The detector destroys all coherence between states of different energies in the laboratory system.

(b) The detector nucleon is confined in the laboratory and not in a momentum eigenstate.

(c) Wave function vanishes outside the laboratory where probability of finding the nucleon vanishes

(d) The wave function is a coherent linear combination of momentum eigenstates which vanish

outside the laboratory.

(e) Neutrino states with the same energy and different momenta δp in the laboratory system can be

absorbed coherently by this detector if δp ¿ L where L is the length of the detector.

5. Relative phases between ν states with same energy and different momenta change with propagation

in space and produce oscillations in detector signal as function of distance from source.

6. Measuring the oscillator wave length gives the values of differences between squares of neutrino masses.

Energies and momenta of states with masses m and m + δm, momenta p and p + δp, same energy E satisfy

E2 = p2 + m2 = (p + δp)2 + (m + δm)2; −2pδp ≈ (m + δm)2 −m2 ≡ ∆(m2)

A pair of components in initial ν state with definite momentum difference can produce the oscillating ν,

31

Dicke Superradiance shows how to measure δp

Darmstadt oscillations observed in decay rate of initial state

Much nonsense written using fancy theory - Follow Racah and look at experiment

Following Racah: What can we learn about the initial state in Darmstadt experiment?

Each member of pair can produce the same final state.

Two components with different momenta interfere and cause oscillations in the transition matrix element.

1. The propagation of the initial state from source to detector is described by the Schroedinger equation.

2. The wave function of the state produced after the decay is completely determined and independent

of whether or not the ν is detected.

3. The decay transition is a first order weak interaction described by the Fermi Golden Rule.

4. The transition operator is invariant under translation in the laboratory system, therefore momentum

is conserved in the laboratory system.

5. The states with the same energy and different momenta that produce the ν oscillations after the

transition must be present in the initial state.

6. Energy is not conserved in the transition because of energy-time uncertainty.

(a) The final states produced in the decay of a one-particle state have finite width.

(b) At short times the width is much broader than the natural line width.

7. Initial states with slightly different energies can produce a final state with a single energy at times

where their broadened widths overlap.

32

What can we know about the initial wave function in the Daramstadt experiment?

1. Two coherent states with a definite momentum difference δP must appear in the final state

2. Produced from two coherent components with the same momentum difference δP in the initial state

3. Two coherent components of the final state that produce oscillations have the same energy

4. A one-particle initial state containing two coherent components with the momentum difference δP

5. Has a definite energy difference δE between the two components. This violates energy conservation.

6. The relative phase between these two components will vary with time as δE · t7. The transition probability will oscillate with a frequency δE.

The period of oscillation δt of the transition probability is then obtained for any experiment

which measures the time dependence of the transition probability; e.g. the Darmstadt experiment .

For two contributing neutrinos whose mass squared difference is ∆(m2) a very crude approximation gives

δt ≈ 2πδE = 4πγM

∆(m2) ; ∆(m2) =[

4πγMδt

]≈ 2.22× 10−4eV2 = 2.75∆(m2)exp

where δt = 7 seconds and ∆(m2)exp are obtained from GSI and KAMLAND neutrino oscillation experiments

γ is Lorentz factor (E/M),

This crude approximation assumes the initial state is a free particle, uses simple relativistic kinematics

to relate the energy E and mass M of the initial state to ∆(m2) and neglects the effect on the energy of the

initial state from its motion through electromagnetic fields in the real experiment,

That this value for ∆(m2) differs by less than a factor of three from the experimental ν masses suggests

that better calculations for the motion theough the storage ring can give a precise result.

33

Dicke Superradiance shows how to measure δp

1. Experiment starts with initial state containing pairs of components with just the right momentum

difference δp needed to produce the final electron neutrino.

2. Dicke has shown that whenever several states can decay into the same final state a particular linear

combination of these states can be called a “superradiant” state.

3. In this state all the components interfere constructively and enhance the transition.

4. The other states called “subradiant” have suppressed transitions.

5. In the GSI experiment the initial state of the radioactive ion is a linear combination of pairs of states

in which each component can produce the same final state.

6. In each pair one can define superradiant and subradiant linear combinations having enhanced and

suppressed decays.

7. Pairs with momentum difference δp have energy difference δE

8. Two components with different energies δE travel around a storage ring with a relative phase difference

δE · t which varies linearly with time.

9. States with this time dependent phase difference oscillate between superradiant and subradiant states

which have well defined relative phases.

10. Periodic oscillations arise and produce observed periodic modulation in decay rate.

34

Dicke Superradiance explains Darmstadt oscillations

Radioactive ion circulates in storage ring and decays by K-capture.

1. Monitoring at regular time intervals collapses wave function

2. Decay time not measured, only disappearance of ion from orbit

3. Time of last monitoring before disappearance is not exponential in time

4. Period of sinusoidal modulation related to neutrino mass differences

Momentum is conserved in weak transition between initial state to final state

1. Coherent components of final state with same energy and different momenta

2. Produced by components in initial state with same momentum difference δp

3. One-particle states with different momenta have different energies; energy conservation is violated

(a) Decaying states do not have a definite energy; natural line width

(b) Width is broadened at times much shorter than lifetime

(c) Broadened widths from decays of states with different energies can overlap and interfere


1. Transition matrix for decay of any linear combination depends upon relative phases

2. Superradiant state with phases adjusted for maximum constructive interference

3. Subradiant states are orthogonal to wuperradiant state.

4. relative phases between components with sifferent energies change in time

Oscillations are transitions between Dicke superradiant and subradiant states.

35

The time dependence of the decay process

The decay process is described in quantum mechanics by the equation

|i(t)〉 = eiHt |i(0)〉|i(t)〉 denotes the wave function of the system at time t and H denotes the hamiltonian.

Both |i(t)〉 and H are unknown at the beginning of the analysis of the observed experimental data and

in the predictions of future experiments.

ν entering detector linear combination of states with different neutrino masses, energies and momenta.

The absorption process is a “which path” or “two slit” experiment in momentum space.

Coherence and interference only if we cannot know through which slit the particle passed.

In absorption of different components of ν wave function only if we cannot determine which component was

absorbed.

Components with the same energy and different momenta are coherent.

Wave function must show vanishing probability for finding nucleon outside detector for all times.

Contains states with the same energy and different momenta with relative phases adjusted to make it

vanish outside the detector.

Two components of ν wave function with slightly different momenta and same energy enter the detector.

Both can turn into same final state containing the same given lepton state.

Transition is coherent if both induce a transition of the detector nucleon to any given final state which

also has a vanishing probability of being found outside the detector.

36

Relative phase of the two neutrino components determines the overlap integral between the initial and

final states of the detector and the transition probability.

Impossible in principle to determine which momentum component was absorbed by detector nucleon.

Components with different energies have random relative phases and are not coherent.

Any interference between states with different energies produce oscillations in time.

Probability of finding nucleon outside the detector must vanish for all times and cannot oscillate in time.

Impossible to obtain any information from the relative phases of nucleon states with different energies.

Energy of the detector nucleon that absorbs the ν can be measured before and after the absorption.

We can know which path in energy-space was taken by the ν and there is no coherence.

Does not matter whether the energy measurable in reasonable experiment; Measurement possible in principle,

Neutrino oscillations can only be produced by interference between components of the neutrino wave

function with the same energy, different masses and different momenta.

Subtle question involving quantum mechanics and theory of measurement

In any neutrino oscillation experiment the neutrino that is emitted after decay has a well defined wave

function which was determined by the previous history of the decaying particle and quantum mechanics.

We may not know the wave function but assume that such a wave function exists

Same wave function present in any repetition of the experiment, whether ν is detected or not.

The detection can project out some features of the wave function.

Cannot change wave function before decay that was created by quantum mechanics and previous history.

37

What we can learn from experiment and theory

What we learn from neutrino oscillation experiments

1. Neutrino oscillations are observed in detectors in experiments as a function of the distance between

the source and detector.

2. The neutrino emitted from the decay is a coherent mixture of states with different four-momenta.

3. The neutrinos are detected in a large many-body system with finite spatial dimensions and effecively

infinite mass in thermal equilibrium with an environment at rest in the laboratory system.

4. The detector is a quantum-mechanical system with definite energy levels.

(a) The detector destroys all coherence between states of different energies in the laboratory system.

(b) The detector nucleon is confined in the laboratory and not in a momentum eigenstate.

(c) Wave function vanishes outside the laboratory where probability of finding the nucleon vanishes

(d) The wave function is a coherent linear combination of momentum eigenstates which vanish

outside the laboratory.

(e) Neutrino states with the same energy and different momenta δp in the laboratory system can be

absorbed coherently by this detector if δp ¿ L where L is the length of the detector.

5. The relative phases between ν states with the same energy and different momenta changes with the

propagation in space and produces oscillations in the signal at the detector as a function of the

distance from the source.

6. Measuring the oscillator wave length gives the values of differences between squares of neutrino masses.

38

The energies and momenta of two ν states with masses m and m + δm, momenta p and p + δp and the

same energy E satisfy the equations

E2 = p2 + m2 = (p + δp)2 + (m + δm)2; −2pδp ≈ (m + δm)2 −m2 ≡ ∆(m2)

Measuring the oscillation wave length gives the squared mass difference ∆(m2) between the two ν states.

Enough information about ν state entering detector to allow conclusions about ν mass differences

No further information needed about the initial state and transition matrix element to obtain ∆(m2)

39

What we learn from weak interaction theory about the decay process

Neutrino oscillations observed in a detector that destroys all coherence between states of different energies.

Oscillations show the neutrino wave function leaving the decay is linear combination of pairs of states

with same ν energy and different masses and momenta with just the relative magnitude and phase to produce

oscillations.

No information about relative magnitudes and phases of components of wave function with different

energies.

This information is destroyed by the detector.

We now know the ν wave function emitted after the decay has pairs of states with same energy and

different neutrino masses and different momenta and the right relative phase to produce oscillations.

This property of the wave function was produced by its previous history and cannot be changed by any

future interaction following the decay.

We also know that momentum is conserved in the transition. Therefore there must be components in

the initial state wave function that have the same momentum difference.

40

A simple example of resolution of the paradox

Time dependent perturbation theory shows violation of energy conservation by energy-time uncertainty

in sufficiently short times. The time dependent amplitude βf (Ei) for the decay from an initial state with

energy Ei into a final state with a slightly different energy Ef is

βf (Ei)g

· (Ei − Ef ) =[e−i(Ei−Ef )t − 1]

]· e−2iEf t

where we have set h = 1 and g is the interaction coupling constant.

We now generalize this expression to the case where two initial states with energies Ei − δ and Ei + δ

decay into the same final state with energy Ef and define x ≡ Ei − Ef

e2iEf t

g· [βf (Ef + x− δ) + βf (Ef + x + δ)] =

[e−i(x−δ)t − 1

(x− δ)

]+

[e−i(x+δ)t − 1

(x + δ)

]

The square of the transition amplitude denoted by T is then given by

|T 2|g2

≡[βf (Ef + x− δ) + βf (Ef + x + δ)

g

]2

= 4 ·[sin2[(x− δ)t/2]

(x− δ)2+

sin2[(x + δ)t/2](x + δ)2

]+ Tint

where the interference term Tint is

41

Tint =[e−i(x−δ)t − 1

(x− δ)

]·[ei(x+δ)t − 1

(x + δ)

]+ cc = 4

[2 sin2[δt/2] + 2 sin2[xt/2] cos[δt]− sin2(δt)x2 − δ2

]

If the time is sufficiently short so that the degree of energy violation denoted by x is much larger than

the energy difference δ between the two initial states, x À δ and

x À δ; |T 2| ≈ 8g2 ·[sin2[xt/2]

x2

]· [1 + cos δt]

The transition probability is given by the Fermi Golden Rule. We integrate the the square of the

transition amplitude over Ei or x, introduce the density of final states ρ(Ef ) and and assume that δ is

neglibly small in the integrals.

+∞∫

−∞|T 2|ρ(Ef )dx ≈

+∞∫

−∞8g2 ·

[sin2[xt/2]

x2

]· [1 + cos δt]ρ(Ef )dx

The transition probability per unit time W is then

W ≈ 4g2 ·+∞∫

−∞du

[sin2 u

u2

]· ρ(Ef )[1 + cos(δt)] = 2πg2ρ(Ef )[1 + cos(δt)]

When energy uncertainty is larger than energy difference between two initial states. cos(δt) ≈ 1

42

Interference term between two initial states comparable to the direct terms

This example shows in principle how two initial states with a given momentum difference can produce

a coherent final state containing two neutrinos with the same energy and the given momentum difference.

A measurement of the momentum difference between the two initial states can provide information on

neutrino masses without detecting the neutrino.

In this simple example the amplitudes and the coupling constant g are assumed to be real. In a more

realistic case there is an additional extra relative phase between the two terms which depends upon the

initial state wave function.

In the GSI experiment this phase varies linearly with the time of motion of the initial ion through the

storage ring.

This phase variation can produce the observed oscillations.

43

The Darmstadt Experiment - Can neutrino mass information be obtained without detecting the ν?

Oscillation observed in radioactive ion decay with emission of unobserved neutrino.

New and very interesting method for determining neutrino masses and mixing angles

When Paul Kienle called my attention to this experiment in December 2006

I said that it was nonsense, inconsistent with causality and quantum mechanics

How can a radioactive ion know about neutrino masses BEFORE it decays?

After some serious thinking and discussions with Paul and Walter Henning

I began to think that there may be some sense in this experiment.

They told me to contact Fritz Bosch. This led to many discussions

With Paul Kienle, Walter Henning, Fritz Bosch, Yuri Litvinov and Andrei Ivanov

Many thanks!!!!!

My approach differs from their theoretical derivation; I start from the end

If we know the neutrino mass, there are no oscillations.

Why is this not a “missing mass” experiment?

No oscillations if conservation laws determine mass of unobserved neutrino

We start with a well-defined initial state of a radioactive ion

Why can’t conservation laws for the decay to final state determine neutrino mass?

What prevents us from knowing the neutrino mass?

44

Can ν masses be measured without observing the ν?

K-capture decay of radioactive atom moving in storage ring

Decay in time with emission of νe not exponential. Ten second modulation

Related to the mass difference between ν eigenstates in νe wave function

How can a radioactive nucleus know about ν masses

BEFORE EMISSION OF AN UNOBSERVED ν?

Essential quantum mechanics

Initial state of radioactive “Mother” ion decays into another “Daughter” ion

Emission of νe - linear combination of several ν mass eigenstates.

If initial state has definite momentum and energy

Conservation of energy and momentum determine ν mass

In this “Missing mass” experiment; ν mass determined without ν observation

Interference between amplitudes from different ν mass states not observable

Interference is possible only if we can’t know everything

Ignorance alone does not produce interference; Quantum mechanics must hide information

What do we know? what can we know? what can’t we know?

Energy-time uncertainty - Violation of energy conservation is crucial

Enables nonconservation of energy in sufficiently short transition time

45

Can neutrino masses be measured without detecting the neutrino?

A standard missing mass experiment determines the mass of an unobserved particle

Conservation laws determine the mass

Neutrino oscillations are produced by coherent mixture of different mass eigenstates

Coherent mixed-mass neutrino states are not produced in a missing mass experiment

Something prevents knowing neutrino mass from conservation laws

Is this a “missing-two-mass” experiment? What do we know? What can’t we know?

1. Momentum is conserved in the transition

2. Energy-time uncertainty prevents use of energy conservation

3. Final state state has coherent pairs of states with same energy; different momenta

4. Initial state can contain coherent pairs with same momentum difference

5. Initial state is one-particle state; different momenta means different energies

6. Two components of initial state can produce the same final state

7. Dicke superradiance whenever two components can produce same final state.

(a) Subradiant state is linear combination whose amplitudes cancel in final state

(b) State orthogonal to subradiant state is called superradiant.

(c) State before transition is combination of superradiant and subradiant states

(d) Momentum eigenstates have different energies, relative phase changes with time.

(e) Initial state oscillates in time between superradiance and subradiance

46

centenary of giulio racah jerusalem 2010 guidance by …in 1950 all particle physicists knew that...

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