IPMU12-0189

Using Neutrinos to test the Time-Energy Uncertainty
Relation in an Extreme Regime

Ramaswamy S. Raghavan,a Djordje Minic,a Tatsu Takeuchi,a,b and Chia Hsiung TzeaabCenter for Neutrino Physics, Department of Physics, Virginia Tech, Blacksburg, VA 24061, USA
Kavli Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwashi, Chiba-ken 277-8568, Japan

E-mail: dminic@vt.edu, takeuchi@vt.edu, kahong@vt.edu
Abstract: We discuss a direct test of the relation of time and energy in the very longlived decay of tritium (31 H) (meanlife τ ∼ 18 yrs) with the width Γ ∼ 10−24 eV [set by
the time-energy uncertainty (TEU)], using the newfound possibility of resonance reactions
3 H ↔ 3 He with ∆E/E ∼ 5 × 10−29 . The TEU is a keystone of quantum mechanics, but
12
probed for the first time in this extreme time-energy regime. Forestalling an apparent
deviation from the TEU, we discuss the ramifications and a possible generalization of the
TEU as ∆E ∆t ≥ (~/2)[1 + (∆t/T∗ )n ] where ∆t = τ is the time of measurement (thelifetime of the state), T∗ = L∗ /c the time for light to cross the Universe ∼ 3 × 1018 s, andn a parameter subject to future measurements. (by R. S. Raghavan.)Keywords: Mössbauer neutrino, time-energy uncertainty relation
ArXiv ePrint: 1210.XXXX

This paper is dedicated to the memory of Raju Raghavan,
a dear colleague and a wonderfully creative physicist.

1

Introduction

In this note, we collect some essential thoughts from an unfinished collaboration with ourdear colleague Ramaswamy (Raju) S. Raghavan, concerning the role of neutrino probes
in the foundations of quantum theory and in the search for the elusive quantum theoryof gravity. This project was interrupted by Raju’s untimely passing on October 20, 2011.
The current paper is motivated by the occasion of the Virginia Tech Symposium on theLife and Science of Dr. Raju Raghavan (October 20, 2012).
The starting point of our collaboration was a proposal by Raghavan [1–6] to useMössbauer neutrinos to check the time-energy uncertainty (TEU) relation ∆t ∆E ≥ ~/2 inthe regime of extremely large uncertainty in time: ∆t ∼ 109 s, and correspondingly small
uncertainty in energy: ∆E ∼ ~/∆t ∼ 10−43 J ∼ 10−24 eV. His idea was to perform aneutrino version of the famous experiment by Wu et al. from 1960 [7–9] which utilized thethen recently discovered Mössbauer effect in γ-rays [10–13].
It had been suggested by Mead (1966) [14, 15] that possible manifestations of a funp
damental length scale, such as the Planck scale `P = G~/c3 , could be seen in this small∆E regime as deviations of ∆E from the expected ∼ ~/∆t behavior. We will discuss thisclaim in more detail later. Following Mead, Andryushin and Melnikov (1973) [16] proposed the idea of using the Mössbauer effect to measure the width of the resonant reaction
3 H ↔ 3 He + ν̄ to extreme accuracy to look for such a deviation. Kells and Schiffer (1984)e1
2[17–19] discussed the necessary conditions for such an experiment to be implemented. To
this, Raghavan brought in the bold proposal of using Niobium tritide as the actual sourceof recoilless emission and absorption of the neutrinos.
Raghavan’s proposal lead to a couple of intense controversies, one on whether theMössbauer effect could truly be realized using Niobium tritide [5, 20, 22–25], and anotheron the interpretation of the Time-Energy uncertainty relation and its implications for the
proposed experiments [26–33]. At the same time, it generated a great deal of excitement
and interest on the new kinds of experiments that would be made possible if Mössbauer
neutrinos became a reality, and the types of physics it will allow us to probe [34–37]. For
instance, it would be possible to perform neutrino oscillation experiments on a table-top
scale [1, 2]. We could also redo all the Mössbauer effect based γ-ray experiments withneutrinos, providing us with a non-electromagnetic probe into nature’s workings.
The three of us (Minic, Takeuchi, Tze) came into the picture as theorists interested
in understanding quantum gravity, and how the existence of a fundamental length scale
may manifest itself in nature. The task Raghavan charged us with was: in the event that
Raghavan’s experiment was performed and a deviation from the ∆E ∼ ~/∆t behavior is
discovered, what would it tell us about the nature of space-time and quantum gravity? In
this paper, we would like to present some of our thoughts on this subject.
But before we do that, we would like to begin with some of the background andarguments surrounding Raghavan’s proposal, and what all the fuss was about. This, we
do in section 2 where we review the history and controversies on the idea of Mössbauer
neutrinos, and those on the Time-Energy uncertainty relation. Section 3 is based on a–2–

draft written by Raghavan dated March 23, 2010, in preparation for a paper the four of
us would have written together. In it, he discusses his main ideas and claims on what
his experiment could have achieved. We have retained his original words as much as
possible, except for a few corrections of typographical errors and changes of some symbols
for notational consistency. Footnotes have been added where we felt some clarification was
called for. Section 4 is based on our contributions during the many meetings the three ofus had with Raghavan between Fall 2009 and Spring 2011, in which he poked our brains
for possible theoretical ways that the canonical time-energy uncertainty relation could beviolated. In section 5, we conclude with some comments on Raghavan’s legacy.

2

Background2.1

Mössbauer Neutrinos

The idea of enhancing the neutrino detection cross section via resonant recoilless neutrino
emission and absorption, i.e. the Mössbauer effect (ME), is not new. Almost immediately
after the discovery of the ME in γ-rays (1958) [10–13], Visscher proposed to use monochromatic neutrinos from an electron capture process to achieve a similar effect (1959) [38].1
The idea was further developed by Kells and Schiffer in the early 1980’s [17–19] where theyalso considered the possibility of obtaining monochromatic (anti)neutrinos from boundstate β-decay.
Bound-state β-decay refers to the process in which the electron emitted by a β-decaying
nucleus is captured by the daughter nucleus instead of being emitted into an unboundcontinuum state. For instance, the electron from tritium β-decay could be captured in the
K-shell of the daughter 32 He nucleus leading effectively to a two-body decay:3
1 H (1s)→3
2
2 He (1s ) + ν̄e (18.6 keV) .

(2.1)

Originally proposed as a possible β-decay process by Daudel et al. in 1947 [39–41], thetheory of bound-state β-decay was developed later by Sherk (1959) [42] and Bahcall (1961)[43]. Bahcall calculated the ratio of bound to continuum decay rates in atomic tritium andfoundΓBound /ΓContinuum = 0.0069 = 0.69 % ,(2.2)
of which 22% was attributed to electron absorption by excited states. So the fraction to theatomic ground state of 32 He is 0.54%, which is the number cited by Raghavan in Refs. [2, 3].
It should be noted that the above bound-state β-decay process for tritium has neverbeen observed experimentally. This is not surprising given the difficulty in observing theemitted ν̄e , which is what Raghavan was trying to achieve via the ME. It has been suggested
that one may be able to see this process in an ion accelerator [45], but as far as we areaware, the experiment is yet to be carried out. In fact, the only experimentally detected66+bound-state β-decay process so far is that of 163
[46].
66 Dy
The only evidence that bound-state β-decay exists for tritium is indirect: Budick (1983)[47] noted that the various measurements of the tritium lifetime differed systematically
1

Visscher in Ref. [38] credits D. Nagle for the original idea of using the ME for neutrino detection.

–3–

depending on the environment that the decaying tritium was placed in. Since both thebound-state and continuum decay rates of tritium depend on the electronic environmentof the nucleus [43, 44], Budick calculated the decay rates of atomic 31 H, the ions 31 H+
and 31 H− , and molecular 31 H and found that the continuum rates differed little while thedifferences in the bound-state rates led to a qualitative agreement with the different lifetime
measurements.2
Assuming that bound-state β-decay of tritium to the ground state of 32 He exists andoccurs at the half-a-percent level, the reverse process can occur resonantly if no energy islost to nuclear recoil in either the emission or absorption of the ν̄e . The ME achieves thisby embedding the nuclei in a crystal lattice, and allowing the entire macroscopic lattice toabsorb the recoil momentum (but no recoil energy). While simple to say in words, the actual
physical implementation of such an effect is difficult due to the conflicting requirements for
the ME to occur. On the one hand, both the emitter (31 H) and absorber (32 He) nuclei must
be bound and fixed to the crystal lattice for recoilless emission/absorption to occur. On
the other hand, the same interactions responsible for this tend to broaden the width of the
decay and attenuate the resonance. Doppler effects due to temperature can also lead towidth broadening, and gravitational potentials will detune the resonance.3 The different
chemical properties of 31 H and 32 He, the latter being a noble gas, also makes it difficult
to achieve the exact same environment for the emitter and absorber: 31 H forms chemicalbonds while 32 He does not.
In a series of papers starting in 2005 [1–6], Raghavan argued that the ME could
be achieved by taking advantage of the fact that tritium forms interstitial compounds,
aka “tritides,” with metals, in which the tritium atoms are trapped inside the grid of themetallic crystal [48].4 Raghavan proposed Niobium tritide as the most promising candidate.
By allowing the tritium in Niobium tritide to β-decay, one could replace them with 32 He
which would be distributed in the same environment in the same way as the tritium. He
argued that due to the very long lifetime of tritium [49], the effects of lattice vibrations
would cancel out and not lead to width broadening, and result in the tritium decay being
detected with its very narrow natural width of Γ ∼ 10−24 eV [3–6] and the cross section
enhanced to σ ∼ 10−17 cm2 on resonance.
Raghavan’s assessment of the effect of lattice vibrations had been contested by several
groups who argued that the lattice vibrations do indeed lead to width broadening and an
attenuation of the resonance strength [20, 22–24]. Raghavan was still quite confident of his
result, however, as you can judge from his tone in section 3. Another objection came from
Suzuki et al. [25] who pointed out that with the tritium densely populating the Niobium
lattice, its 1s orbitals would form a band structure, again broadening the resonance width.
Due to these possible difficulties, Potzel [24] has recently argued for the following
163
Holmium163 (16367 Ho) ↔ Dysprosium163 ( 66 Dy) pair as an alternative Mössbauer neutrino
2

Sherk had already noted that the lifetimes of atomic and molecular tritium should be different in
Ref. [42] (1959).
3
The ME was how gravitational red shift was first measured, after all.
4
The study of embedding hydrogen atoms inside metallic crystals has been pursued as a technology forhydrogen storage.

–4–

system:163
11
67 Ho (4f )

↔

163
10
66 Dy (4f ) + νe ,(Q = 2.6 keV) .

(2.3)

This system was also listed by Kells and Schiffer [19] as a possible candidate. Atomic
163 Dy is stable. However, when stripped of all of its electrons it can β-decay into 163 Ho.
66
67
Curiously, this was the first reaction in which bound-state β-decay,
163
66+
66 Dy

→

163
66+
+ ν̄e ,
67 Ho

(2.4)

was experimentally confirmed at the GSI in 1992 [46].
2.2

The Time-Energy Uncertainty Relation

Uncertainty relations play an important role in the quantum theory. They are based on
fundamental general properties of the theory and manifest the nature of it. There are twodifferent types of the uncertainty relations in the quantum theory: Heisenberg uncertainty
relations and time-energy uncertainty relations.
The Heisenberg uncertainty relations are based on commutation relations for hermitian
operators, which correspond to physical quantities. Let us consider two hermitian operatorsÂ and B̂. From the Cauchy-Schwarz inequality we have1hψ| [Â, B̂] |ψi ,
2

(2.5)

q
(∆A)ψ =
hψ|Â2 |ψi − hψ|Â|ψi2(2.6)(∆A)ψ (∆B)ψ ≥where |ψi is some state, andis the standard deviation of the probability distribution of possible outcomes of a measurement of Â on the state |ψi.5 For the canonical pair x̂ and p̂, we have [x̂, p̂] = i~, and thelower bound will be independent of the state |ψi:∆x ∆p ≥

~
.
2

(2.7)

∆t ∆E ≥

~
2

(2.8)

The time-energy uncertainty relation

has a completely different character. While deduced immediately from Eq. (2.7) by demanding relativistic covariance, it does not share the same operator-based foundation, thetime t being a parameter rather than an operator. Consequently, it has been a subjectof intense discussions and controversy from the early years of the quantum theory. In theliterature exist many different derivations of Eq. (2.8), each with different meanings of thequantities that enter into them [52–58].
5

Note that the ‘uncertainty’ (∆A)ψ in the measurement of Â on state |ψi has nothing to do with the‘disturbance’ of the state |ψi by the measurement process itself as originally formulated by Heisenberg in his
famous Heisenberg microscope thought experiment [50]. See Ref. [51] for a clarification of this distinction.

–5–

For instance, Mandelstam and Tamm [52] start from the Heisenberg equation of motioni~dÂ(t) = [ Â(t), Ĥ ] ,
dt

(2.9)

to which an application of Eq. (2.5) yields(∆A)ψ (∆E)ψ ≥~ dhψ|Â(t)|ψi .
2 dt(2.10)

Defining(∆t)ψ ≡(∆A)ψ
dhψ|Â(t)|ψidt,(2.11)

they obtain (∆t)ψ (∆E)ψ ≥ ~/2 with a very particular meaning of the uncertainty (∆t)ψ :it is the time interval which is necessary for the expectation value hψ|Â(t)|ψi to be changed
by one standard deviation (∆A)ψ . In other words (∆t)ψ characterizes the time interval
during which the state of the system significantly varies.
The most straightforward derivation starts from the Schrödinger equation itself [55]i~dψ(t) = Ĥψ(t) ,dt(2.12)

which associates the eigenvalues of the Hamiltonian Ĥ with the oscillational frequency of
the wave-function ψ(t). The relation ∆t ∆E ≥ ~/2 is then the standard relation between
the spread in time and the spread in frequency for any wave phenomenon. Indeed, in thecontext of quantum field theory (QFT), in which the position x is also a parameter liket and not an operator, the relation ∆x ∆p ≥ ~/2 is also the standard relation between
the spread in space and the spread in wavenumber for any wave phenomenon, and both
uncertainty relations are placed on an equal footing.
The uncertainty in energy of an eigenstate of Ĥ created at t = 0 would not be zeroat any finite time t = ∆t since no wave of finite duration can have a definite frequency.
Rather, its frequency will have a spread of ∆E ∼ ~/∆t which can only go to zero in thelimit t = ∆t → ∞. Thus, the ‘uncertainty’ ∆t should be interpreted as the duration of
the state since its creation. If the state is unstable with lifetime τ , then as t → ∞ theuncertainty in the frequency can be expected to settle to its ‘natural’ value of ∆Enat ∼ ~/τ .
This was the phenomenon observed by Wu et al. [7], Lynch et al. [8] and others utilizing
the ME in γ-rays, and which Raghavan proposed to observe using the ME in neutrinos.

3

Experimental Test of the Time-Energy Uncertainty Relation withMössbauer Neutrinos – by R. S. Raghavan

One of the most basic predictions of quantum mechanics (QM) is the time-energy uncertainty (TEU) relation ∆E ∆t ≥ ~/2, in parallel to the Heisenberg relation ∆p ∆x ≥ ~/2.
Since time is not a canonical variable like p̂ and x̂ for writing out a commutator that underlies the uncertainty product, the meaning of the TEU has been debated for decades, even–6–though it has been verified in numerous atomic, nuclear and particle decays – the contexts
for the role of the TEU which sets the minimum observable energy width ∆E = ~/τ . Theenergy-time relation has been tested in numerous experiments [59]. In this note we suggest
a neutrino analog of the famous test of the energy-time relation performed in [7].
It is not clear however, whether the TEU still holds in the limit of extremely small
widths of very long lived states where other factors may enter in altering the minimum
observable energy. For example, Mead [15] has argued that the ultimate measurable enp
G~/c3 (G =ergy width could be limited by the fundamental (Planck) length `P =Newton’s constant) ahead of that expected from the TEU (via an independently measured
state lifetime). He suggests that such an effect – a Planck broadening of spectral lines – may
be observable with energy precision ∆E/E ∼ 10−20 to 10−40 . If there is an experimental
tool to probe this extreme regime (recent work suggests the possibility), the question arises:
If it is found that the Planck limited width is broader than the QM width, what is the roleof time (i.e. the measured lifetime) and the TEU itself which would be in conflict withthe result? This is the question addressed in this paper. We discuss its deep ramifications.
We offer a generalization of the TEU as ∆E ∆t ≥ (~/2)[1 + (∆t/T∗ )n ] where ∆t is thetime of measurement (in this case the lifetime of the state τ , in the present case ∼ 109 s),T∗ ∼ L∗ /c the time for light to cross the Universe ∼ 3 × 1018 s, and n is a parameter
subject to adjustment with future measurements.6 Note that this generalization preserves
the traditional QM foundations7 but introduces parameters that can be connected in the
future to the Planck length and/or quantum gravity.
Recently, however, a precision of ∆E/E ∼ 5 × 10−29 appears possible by the newfoundneutrino resonance reaction 31 H ↔ 32 He (tritium 31 H: lifetime τ ∼ 18 yrs [49], TEU width
Γ(TEU) ∼ ~/τ ∼ 10−24 eV), which offers a meaningful TEU test in an extreme energy-time
regime not probed up to now. The purpose of this note is to discuss first, a direct test of
TEU in tritium decay. Looking ahead to the possibility of a deviation from the quantum
theoretical TEU as conjectured by Mead, we explore generalization of the TEU relation
on foundations beyond quantum theory and their connections to the fundamental length`P and quantum gravity. While generalized forms of ∆p ∆x ≥ ~/2 have been discussed in
the context of quantum gravity (see [60–69] and references therein), we are aware of fewersuch attempts [70] regarding the TEU.
The ν̄e resonance reaction in the case of 2-body β-decay of 31 H, which results in emissionof a mono-energetic 18.6 keV neutrino (ν̄e ), is reminiscent of the well-known γ-ray resonance[Mössbauer Effect (ME)]. But the new aspect is the long lifetime of 31 H which presents
entirely new facets well beyond those of classical ME [3]. The significant and surprising
discovery [3] is that, in this case, a useful fraction of the ν̄e will always be emitted with
the natural width, ~/τ ∼ 10−24 eV in spite of myriad extra-nuclear sources of broadening
and energy shifts well known in the ME work [22]. The reason is that for long lifetimes thefluctuations act to modulate and average out the fluctuations rather than detune the ν̄e
6Raghavan’s idea was that if the large time scale T∗ has its origins in quantum gravity, then it would
be related to the size of the observable universe L∗ . The size of the observable universe is currentlyL∗ ∼ 93 billion light years ∼ 1027 m, which yields T∗ = L∗ /c ∼ 3 × 1018 s.
7
As we will argue in section 4, it will actually rock the very foundations of QM.

–7–

energy. Thus, they affect only the intensity of the resonance signal, not the width as they
do in short lived ME levels [6].8 The maximum resonance cross section σ0 (height of theresonance line) is governed by the flux density of ν̄e emitted within the resonance width∆E. In the case of the natural width ∆Enat , σ0 takes the maximum value of σ0 = 2πλ [22]where λ is the ν̄e wavelength. In the 3 H ↔ 32 He reactions, σ0 is very large, ∼ 10−17 cm2 .
The ν̄e ’s are resonantly absorbed in 32 He, back transforming it to 31 H which then β-decays.
The simplest resonance signal is electron emission of the resonantly induced activity. The
percent absorption at zero detuning of the resonance, i.e. the rate of resonant activation isthe direct experimental handle on the line width. The energy sensitivity of the resonanceis ∆E/E = 10−24 /18.6 × 104 = 5.4 × 10−29 .
The basic questions in the application of the TEU relation ∆E ∆t ≥ ~/2 is the definition of the “time” ∆t in the TEU, the resulting energy width ∆E and the effect on σ0 .9
The roles of these observables have been particularly clarified by the ME experiments by
Wu et al. and others [7, 8], and by theory [9]. The experiments used the classic decay
57 Co → 57 Fe to an initial state at 136 keV in the daughter which decays unmeasurably fast
27
26
to the isomer which emits the well known 14.4 keV ME γ-ray with a mean life of τ ∼ 144 ns.
The experiment detected both the 122 keV precursor and the 14.4 keV photons. The MEof the latter was observed in coincidence with the 122 keV photon. The 122 keV signal
marks the creation of the 14.4 keV state and a known delay T between the two photon
signals measures the duration for which the 14.4 keV state remains unperturbed. The detection of the 14.4 keV photon marks the end of the time of measurement. The resonancesignal detected after this time T measured the precise energy of photon (and the emitting
state) with an uncertainty given by the width of the resonance. The results showed thatthe width depended on T . At short delays T , the width was broad and as T approachedthe lifetime τ of the state the width narrowed to the natural line-width of the state. The
absorption at zero detuning quantitatively measures the resonance width by proxy. Complete agreement with TEU was found in these remarkably clear results. This experimentalso clearly illustrates the definitions of T and E in the TEU. To repeat, the “time” inthe TEU is the unperturbed duration of the unstable state starting from its creation andlasting till the state decays, as detected by its decay product.
This “time-filtered” resonance method can be applied directly without electronic coincidences to the the ν̄e resonance of 31 H ↔ 32 He because of the long lifetime of the 31 H
state. The time starts with the creation of tritium (in a reactor or elsewhere in a timeshort compared to the lifetime of 31 H) and ends with the detection of the resonance signal
3 He → 3 H. It does not start with installation of the 3 H in the experimental set up and2118Footnote by Raghavan: B. Balko et al. (Priv comm. and to be published) have unequivocallyconfirmed theoretically the claims of Raghavan in ref. [3] and [6].
9Footnote by Raghavan: Surprisingly confusing opinions on the definition of “time” have been expressed. Lipkin [26] asserts that the TEU with his definition of the “time” as the time of flight of the ν̄ewould make it impossible to observe the ν̄e resonance. This claim contradicts numerous ME experiments
performed with different baselines with no effect on the resonance. Potzel and Wagner [22] assert that
the TEU does not play any role in the ν̄e resonance line width, while Akhmedov, Kopp, and Lindner [31]conclude that an imprecisely defined “time of the experiment” does. Both claims are refuted by early ME
experiments that address this specific question (see text).

–8–Normalized Max. Absorption

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
100

β=1
ΔEnat

β=1
ΔE = 10ΔEnat

τ=6600 d
1000

10000

100000Radio-Age of Tritium (days)

Figure 1. Growth rates of normalized resonance absorption A with radio-age of 31 H for QM-TEU.
Left curve for ∆E(QM) = ∆Enat , for the measured τ of 31 H for known absorber thickness β = 1.
Right curve for hypothetical ∆E(non-QM) = 10 ∆Enat for β = 1.

starting the counting. It is important to realize that in the TEU framework it is immaterial
whether the created 31 H is stored for a long time before installation in the experiment. The
total time after creation (storage+installation) up to the detection of the neutrino is thedelay time T , thus the resonance signal strength depends on the age of the tritium, a novel
effect demanded by the TEU. By the same token, as the counting proceeds, the delay timeincreases naturally, the effective resonance line-width decreases via the TEU, thus the resonance signal spontaneously increases with time. Observation of these time dependencesis the most straightforward experimental demonstration of the TEU (see Fig. 1).

4

Theoretical Implications - Minic, Takeuchi, Tze

The above discussion was concerned with a new fundamental neutrino probe of the energytime relation in quantum theory. Now we would like to discuss possible theoretical implications of such a new probe of the foundations of quantum theory. Given that the proposed
experiment is sensitive to small deviations from the standard energy-time uncertainty relation∆t ∆E ≥ ~/2 ,(4.1)

one must ask in what way the physics at the Planck scale `P could modify this relation.
First and foremost, in the presence of a fundamental length scale `P one expects spatial
resolutions to be bounded from below ∆x ≥ `P [71]. It has been argued by Mead [15] that
this finiteness in ∆x will naturally lead to an uncertainty in the energy of a particle trappedin a potential V of order∂V∆E ∼ ∆x · (∇V ) & `P
,
(4.2)
∂r

–9–

in addition to any intrinsic energy uncertainty the state may have. If R denotes the characteristic size of a bound system (such as the size of the nucleus), and ν0 the average frequencyof the characteristic transition, then one expects a frequency shift of ∆ν ≥ (`P /R) ν0 . Unfortunately, as long as one works within the confines of canonical quantum theory, one
cannot disentangle the uncertainty due to `P from the intrinsic one. In particular, theenergy-time relation, Eq. (4.1), being a consequence of the Schrödinger equation, remains
unaffected.
Attempts to incorporate the effect of `P into quantum theory has been through thedeformation of the canonical commutation relation between x̂ and p̂ to [60–62][ x̂, p̂ ] = i~ 1 + α x̂2 + β p̂2 + · · ·



,

(4.3)

where the ellipses denote possible higher order terms in x̂ and p̂ which are neglected. This
leads to the relation~∆x ∆p ≥
1 + α(∆x)2 + β(∆p)2 ,(4.4)
2
assuming αhx̂2 i + βhp̂2 i = 0. On dimensional grounds one expects β ∼ `2P /~2 , whileα ∼ L−2∗ where L∗ must be a very large length scale (the size of the universe?) to prevent
obvious phenomenological problems. The appearance of a large scale L∗ from Planck-scale
physics may be justifiable through the UV/IR mixing nature expected of quantum gravity[67]. The α = 0 case is motivated by perturbative string theory [72–77] and leads to√the lower bound ∆x ≥ ~ β ∼ `P . This deformation of [ x̂, p̂ ] can maintain consistency
within canonical non-relativistic quantum theory, and thereby also maintain Eq. (4.1), andvarious phenomenological consequences have been worked out [63–69]. However, if we insist
on rendering Eq. (4.4) covariant, Eq. (4.1) must also be modified to∆t ∆E ≥

~
1 + α(∆t)2 + β(∆E)2 ,
2

(4.5)

which would require a (non-local) modification of the Schrödinger equation [78]. Such ageneralization of the Schrödinger equation may occur in a more fundamental approach toquantum gravity [79–85], but it is conceptually and technically a very challenging task.
As mentioned above, we expect the parameter α to correspond to the inverse square ofsome very long time scale. If we accept Eq. (4.5) purely on phenomenological grounds, α
could be observed, or bounded, in the experiment proposed in this paper, in particular,
the neutrino analog of the Wu et al. experiment [7].
Given the rigidity of canonical quantum theory, a relation such as Eq. (4.5) which
would require a major deformation of the Schrödinger equation itself may be too radical tocontemplate. So let us consider a more conservative option. Suppose that the Hamiltonianof some more fundamental theory, such as a quantum theory of gravity, depends on somelarge fundamental time-scale, T∗ , and that the energy uncertainty is shifted from ∆E to∆E − (~/T∗ ) so that the corrected energy-time uncertainty relation reads∆t∆t ∆E ∼ ~ 1 +.
T∗– 10 –(4.6)

The sign of the shift in ∆E has been chosen to maintain the positivity of the right-handside. This corrected energy-time relation would be reasonable from the point of view of the
proposed experiment for large ∆t. Such a linear shift in time can be accomplished while
preserving unitary evolution by way of a suitable T∗ -dependent operator.
Finally, another possibility is the “inverse” of the preceding modification of the energy
time relationτ∗ ∆t ∆E ∼ ~ 1 +
,
(4.7)
∆t
where τ∗ must be small to prevent phenomenological problems at small ∆t. Such an
expression can be readily obtained from the following local linear generalization of the
Schrödinger equationĤψ = i~ (δt ψ) − i~ τ∗ (δt2 ψ) .
(4.8)
Being diffusive, such a modified Hamiltonian is non-unitary. Moreover, the change fromthe standard energy-time relation, Eq. (4.1), would unfortunately not be observable in theabove proposed experiment, as the fundamental scale τ∗ should be very short.
In the context of the preceeding modification of the Schrödinger equation, it is tempting
to speculate that some new quantum gravitational degrees of freedom [79–85] are responsible for the overall conservation of probability, and that the proper inclusion of these
degrees of freedom would restore a seemingly violated unitary evolution. If indeed quantum gravity is essential for the problem of quantum measurement [86], then the last model,
Eq. (4.7), for the generalized time-energy uncertainty relation presented in this note would
be a manifestation of such fundamentally new physics.

5

Closing Remarks

Raju Raghavan was a remarkably creative physicist. His novel suggestion concerning theMössbauer neutrinos can be used to make neutrinos the next natural probe of the foundations of quantum theory. In this note, we have proposed how neutrinos could be used
to probe the time-energy uncertainty relation which is ultimately related to unitary time
evolution of quantum states. Any departures from this fundamental relation would be indicative of a new fundamental framework for quantum physics, which may be required by
the quantum theory of gravity [89]. Thus, neutrinos can be seen as the next natural probe
not only of the early history of the Universe, via the cosmic neutrino background [90], butalso of the fundamental quantum underpinnings of the Universe itself. We sincerely hopethat future avatars of Raghavan will further develop this fascinating research direction.

Acknowledgments
We wish to thank Patrick Huber and Jonathan Link for useful discussions about the physics
of this proposal. Special thanks go to Bruce Vogelaar for his invaluable help and encouragement. The collaboration of Bohdan Balko with Raju Raghavan is also gratefully acknowledged. This work was supported in part by the U.S. Department of Energy, grant
number DE-FG05-92ER40677, task A (DM and TT) and by the World Premier International Research Center Initiative, MEXT, Japan (TT).

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