The energy-time uncertainty principle and quantum phenomena

For examples of opposing textbook interpretations of the aspects of the energy-time uncertainty principle most relevant to this paper, compare the viewpoint that energy is strictly conserved in quantum mechanics given in Ref. 36, pp. 78–80, to the viewpoint that virtual processes can temporarily violate energy conservation given in

R.

 

Shankar

, , 2nd ed. (, , ), Sec. 9.5. Discussions of aspects of the latter interpretation can be found in Refs. 31 and 65–67. Discussions from a more or less neutral viewpoint are given in Ref. 68.

We assume that virtual processes can temporarily violate energy conservation. In this interpretation energy conservation is absolutely valid in the long term, but the energy-time uncertainty principle is construed as allowing temporary energy fluctuations about a conserved long-term average value.

D. H.

 

Kobe

and

V. C.

 

Aguilera-Navarro

, “,”  , – (), and references cited therein.

The energy-time uncertainty principle is more difficult to interpret than the position-momentum uncertainty principle primarily because position and momentum (and also energy) are operators in quantum mechanics, whereas the status of time in quantum mechanics is a more difficult issue. This point was emphasized by Dr.

D. H.

Kobe

private communication ().

The status of time quantum mechanics, and related issues, are addressed in Ref. 3. Dr. Kobe also pointed out to me that the usual definition of the uncertainty product requires some refinement in order to be completely accurate, even though it is sufficiently accurate for our purposes in this present paper.

See, for example,

J.

Hilgevoord

and

J. B. M.

Uffink

, “,”  , – (); and

J. B. M.

 

Uffink

, and

J.

 

Hilgevoord

, “,”  , – (), which however, considers the position-momentum rather than the energy-time uncertainty principle.

If relativistic effects are important, then “energy” should be interpreted accordingly. See Refs. 69–73.

It has been stated that the relations $E=hν$ and $p=h/λ$ cannot be understood in terms of anything more basic. See

L.

Brillouin

, (, , ), Chaps. 1 and 3, and especially Secs. 3.3 and 3.4, and p. . We do not know why, for example, Planck’s constant $h$ has its measured value, nor why it is $h$ rather than another dimensionally equivalent constant that “binds” $p$ and $λ$⁠, and $E$ and $ν$⁠. (Dimensional equivalence need not imply physical equivalence; for example, entropy and heat capacity are dimensionally but not physically equivalent.) However, Brillouin’s statement may be too strong because the proportionalities $p∝1/λ$ and $E∝ν$ are suggested by special relativity. See Ref. 69, p. 13, Exercise 2.6, p. 58, Sec. 6.8, and Exercise 6.20. See also Ref. 70, pp. 2 and 62–63, Secs. 32–33, and Exercises 24 and 25 of Chap. V. In general relativity the proportionality $E∝ν$ ensures consistency of gravitational time dilation and energy conservation during gravitational frequency shift. See Ref. 69, Sec. 1.16, Chap. 9, Secs. 11.6 and 12.2, and Exercises 1.8–1.11, 6.20, and 9.1–9.4;

An excellent summary of these points is in

P.

Fong

, , (, , ), pp. –;

Also, see relevant references cited therein. A relevant summary of de Broglie’s viewpoints concerning $h$ is in

L.

 

de Broglie

, (, , ), translated by A. J. Pomerans.

P. C. W.

 

Davies

, “,”  , – ().

P. M.

Alsing

and

P. W.

Milonni

, “,”  , – (), and references cited therein.

Also,

V. F.

 

Mukhanov

and

S.

 

Winitzki

, (, , ), p. and Chap. 8.

F.

 

Mukhanov

and

S.

 

Winitzki

, (, , ), pp. – and the solution to Exercise 1.5.

See

Bernard L.

 

Cohen

, “,”  , – (), and the reference cited therein.

H. M.

 

Georgi

, in , edited by

P.

 

Davies

(, , ).

J.

 

Taylor

, in , edited by

P.

 

Davies

(, , ).

R.

 

Adler

, in , edited by

G.

 

Fraser

(, , ).

Michael B.

 

Green

, in , edited by

G.

,

Fraser

(, , ).

Charles W.

 

Misner

,

Kip S.

 

Thorne

, and

John Archibald

 

Wheeler

, (, , ), Sec. 43.4. In this present paper, we can neglect the “extra” term in the generalized uncertainty principle as discussed in Ref. 13, Sec. 2.9.

See

Leonard I.

 

Schiff

, , 3rd ed. (, , ), Sec. 17.

See

Rolf

Landauer

, “,”  , – ();

M.

Büttiker

and

R.

Landauer

, “,”  , – ();

Markus

 

Büttiker

and

Rolf

 

Landauer

, “,”  , – () and references cited therein. These authors often interpret the (nonrelativistic) result $Δt=a[m/2(V−E)]1/2$ as a (nonrelativistic) tunneling particle’s barrier-traversal time as we do. Sometimes these authors interpret $Δt$ as something more like a (nonrelativistic) tunneling particle’s interaction time with the barrier.

Numerous approaches agree with the results in Ref. 10 for the tunneling time, at least for barriers that are not highly transparent. See

Mark J.

Hagmann

, “,”  , – ();

Mark J.

Hagmann

, “,”  , – ();

Mark J.

Hagmann

, “,”  , – ();

Mark J.

Hagmann

, “,”  , – ();

Mark J.

 

Hagmann

, “,”  , – ().

Many approaches agree that the nonrelativistic results in Ref. 10 require barriers that are not highly transparent. See Refs. 17 and 18. Also, reflection from a totally opaque potential step, especially of a wide wave packet, manifests major differences in behavior from tunneling through even a nearly opaque potential barrier. If the second strong inequality in Eq. (12) is reversed as for the case of the time delay upon reflection from an infinitely wide potential step for which $V>E$⁠, and $aefold$ is the $e$-folding penetration distance of $|Ψ|2$ into a wide $(Δx⪢aefold)$ wave packet, then even at midreflection only a small part of the wave packet enters the step; most of it remains outside (as incident and reflected waves). See Refs. 20 and 68.

Claude

 

Cohen-Tannoudji

,

Bernard

 

Diu

, and

Frank

 

Laloë

, (, , ), Complement $JI$⁠.

As discussed in Ref. 20, Complement $HI$⁠, the $e$-folding distance of $|Ψ|2$ is $aefold=ℏ/[8m(V−E)]1/2$ into an infinitely wide potential step for which $V>E=12mv2$⁠; $+v$ $(−v)$ is the particle velocity outside the step before (after) reflection. Because the decay is exponential, $aefold$ is also the average penetration into the step of the small part of a wide $(Δx⪢aefold)$ wave packet that penetrates. The round-trip distance within the step traversed by this part of the wave packet upon reflection is $2aefold$⁠. As shown in Ref. 20, the time delay upon reflection is $τ=(ℏ/v)[2/m(V−E)]1/2$⁠. Hence, for this part of the wave packet, the effective speed within the step during the reflection process is $u∗=2aefold/τ=v/2$⁠. This result for $u∗$ disagrees with our nonrelativistic result in Eq. (6). However reflection from a potential step of infinite width manifests major differences from tunneling through a potential barrier for which $V>E$ of finite width (or possibly of infinitesimal width if infinitely tall). In tunneling the entire wave packet (not just a small part) must traverse the entire barrier. Also as discussed in Ref. 20, Complement $AIII$⁠, if the second strong inequalities in Eq. (10) and/or Eq. (12) are reversed, then the probability distribution of the (nonrelativistic) velocity of a particle of mass $m$ in a kinetic energy eigenstate with eigenvalue $K$ has, during its interaction with a barrier (or well) of width $a$⁠, a large relative dispersion about the rms (classical) value of $(2K/m)1/2$⁠.

See Ref. 10, p. 97,

Robert

Gomer

, (, , ), pp. –,

Phillip H.

 

Bligh

, “,”  , – ().

The signal velocity is defined as “including all points of nonanalyticity, that is, new information that is not already foretold in an earlier portion of the waveform.” It is not “the velocity of propagation of the half-the-peak-intensity point on the leading part of the pulse” (Ref. 31). Even in cases where the phase velocity, the group velocity and/or the energy transport velocity exceed $c$⁠, the signal velocity thus defined does not exceed $c$⁠, and hence causality is not violated regardless of the motion of transmitter and/or receiver. See Ref. 31, Chaps. 1–4;

Raymond Y.

Chiao

,

Paul G.

Kwait

, and

Aephraim M.

Steinberg

, “”  (), – ();

A. M.

Steinberg

,

P. G.

Kwait

, and

R. Y.

Chiao

, “,”  , – ();

Raymond Y.

Chiao

, “,”  , – ().; and

D. R.

Solli

,

C. F.

McCormick

,

C.

Ropers

,

J. J.

Morehead

,

R. Y.

Chiao

, and

J. M.

Hickmann

, “,”  , -1–4 ();

Another viewpoint is in

Edward

 

Gerjuoy

and

Andrew M.

 

Sessler

, “,”  , – ().

See

Chris

 

Isham

, in , edited by

Paul

 

Davies

(, , ). The inverse-square part of this decrease in forces with distance is a purely geometrical consequence of (Euclidean) space being three-dimensional.

The tachyonic regime is discussed in

Gerald

Feinberg

, “,”  (), – ();

G.

Feinberg

, “,”  , – ();

Olexa-Myron

Bilaniuk

and

E. C. George

Sudarshan

, “,”  (), – ();

J. D.

Elder

,

G. L.

Trigg

,

O. M.

Bilaniuk

, and

E. C. G.

Sudarshan

, “,”  (), and 79 (); and

O. M.

 

Bilaniuk

,

S. L.

 

Brown

,

B.

 

DeWitt

,

W. A.

 

Newcomb

,

M.

 

Sachs

,

G.

 

Sudershan

, and

S.

 

Yoshikawa

, “,”  (), – (). See also references cited in these works.

R. P.

 

Feynman

and

A. R.

 

Hibbs

, (, , ).

See Ref. 15, Chap. 43, and Secs. 44.2–44.3. For more recent work see

W. A.

Christiansen

,

Y. Jack

Ng

, and

H.

van Dam

, “,”  , -1–4 ();

W. A.

Christiansen

,

Y. Jack

Ng

, and

H.

van Dam

, “, (E) ();

Also,

Y. Jack

Ng

, “,” in , edited by

G.

Amelino-Camelia

and

J.

Kowalski-Gilkman

(, , );

and

Y. Jack

Ng

and

H.

van Dam

, “,”  , – ();

More qualitative discussions are given in Refs. 13 and 14. A recent review is given in

Lee

Smolin

, “,”   (), – (). In this present paper, we consider mainly distances and time intervals much larger than the Planck dimensions, with only one brief conjecture (in the next-to-last paragraph of Sec. II C 1) concerning hypothetical distances and time intervals smaller than the Planck dimensions. Hence, we can neglect the “extra” term in the uncertainty principle discussed in Ref. 13, Sec. 2.9. An alternative viewpoint of the quantum vacuum is summarized in

C.

Lanczos

, (, , ), pp. –; with a more detailed discussion in

C.

Lanczos

, “,”  , – ().

Recent work suggesting a possible breakdown of special relativity that is related to quantum field theory is reviewed in

Ron

 

Cowen

, “,”   (), ().

Catherine

 

Asaro

, “,”  , – (), and references cited therein.

Nick

 

Herbert

, (, , ), Chap. 7.

See

Gordon

 

Kane

, (, , ). The laws of lepton-number and baryon-number conservation are very good approximations but do allow permanent violations in rare and/or extreme circumstances. In contrast, the law of energy conservation is absolutely valid in the long-time limit $Δt→∞$⁠, even if the energy-time uncertainty principle is interpreted as allowing temporary energy fluctuations about an absolutely conserved long-term average value.

P. W.

 

Milonni

, (, , ), p. , and Secs. 3.1.1, 3.2.1, and 3.8.

Reference 69, Sec. 2.10 (especially the first paragraph). Also, Ref. 70, Item (ix) of Sec. 7 (especially the first paragraph).

Richard C.

 

Tolman

, (, , ), Sec. 52.

Note that the maximum signal velocity consistent with causality, $usignal,max=c2/vrel$ (⁠$vrel$ is the relative velocity between the transmitter and receiver), is also the de Broglie wave speed.

We mention that extrasensory perception if it exists could propagate superluminally as fast as $usignal,max=c2/vrel$ and still be a causal phenomenon. The possibility of some phenomena being supercausal and not merely superluminal if they exist is discussed in , edited by

Daniel P.

Sheehan

(, , ) (By contrast, according to standard special relativity, even such unproven phenomena if they exist cannot propagate faster than $c$⁠, even though this need not violate causality. See Ref. 69, p. 27, and Ref. 70, p. 16.)

For general discussions of both sides of the issue, see for example: Dr.

E.

Mitchell

with

D.

Williams

, , Revised Edition (, , ); and

R.

Matthews

, “,” BBC Knowledge, –, especially p. 77 (May–June ); versus

M.

Kaku

, (, , ), Chaps. 5, 6, and 15, with Notes and references for Chaps. 5 and 6 on pp. –; and

J.

 

Randi

, (, , ).

D. J.

 

Griffiths

, , 2nd ed. (, , ), Sec. 3.5.3.

Reference 69; Sec. 2.10 (especially the seventh, eighth, and ninth paragraphs), Sec. 2.11, and the references cited therein; and Exercises 2.6, 2.7, and 2.12 on pp. 58–59. Also, Ref. 70, Sec. 7 [especially the first, third, and fourth paragraphs of Item (ix)] and Exercises 5–7 and 15 of Chap. I on pp. 21–23.

The possibility of superluminal but not supercausal signals is discussed in

Mitchell J.

Feigenbaum

, “,” arXiv:0806.1234v1,

and reviewed by

Mark

 

Buchanan

, “,”   (2680), – ().

See Refs. 11 and 12.

Reference 69, Sec. 3.6 (especially the last paragraph). Also, Ref. 70, p. 18, and Exercise 5 of Chap. II on p. 36.

Reference 69, Secs. 2.9–2.11 (especially Fig. 2.6 on p. 50, the first complete paragraph on p. 51, and the third complete paragraph on p. 56) and Exercises 2.6 and 2.7 on pp. 58–59. Also, Ref. 70, p. 18, Exercises 6, 7, and 15 of Chap. I and Sec. 32.

An alternative qualitative description of the thermal nature of Unruh radiation is given in

Lee

 

Smolin

, (, , ), Chap. 6. In Chap. 7 this description is extended to Hawking radiation in accordance with the equivalence principle.

Transition probabilities are discussed in Ref. 20, Chap. XIII (especially Sec. C.3.b). See also Ref. 67, Sec. 10.1.

See

S. W.

Hawking

, “,”  , – ();

S. W.

Hawking

, “,”  (), – () and references cited therein.

See also

F.

 

Mukhanov

and

S.

 

Winitzki

, (, , ) and Ref. 13.

A qualitative discussion is given in Ref. 42.

Reference. 69, p. 8. In a few well-chosen words: “$…$ while in general relativity all matter, including its motion, undoubtedly affects local inertial behavior, it appears not entirely to cause it.”

Reference 69; Exercise 2.1 on p. 58; Sec. 4.4; Exercises 4.5, 4.6, 4.10, 4.13, 4.14, and 4.16 (especially Exercise 4.10); and p. 104. Also, Ref. 70; pp. 1, 42–43, and 65; and Exercises 9, 6, and 17 of Chap. III; and Ref. 33, Sec. 55.

J. J.

 

Sakurai

, (, , ), pp. –.

Transparency can be induced even if $n$ is considerably greater than one. See

Marlan O.

 

Scully

and

M.

 

Suhail Zubary

, (, , ), pp. – and Secs. 7.3 and 7.6.

Richard P.

 

Feynman

,

Robert B.

 

Leighton

, and

Matthew

 

Sands

, (, , ), Vol. , Secs. 42–5.

See Refs. 1–5.

See, for example, Ref. 20, Sec. D.2.e of Chap. III; Ref. 65, pp. 78–80 and Secs. 5.6 and 5.8; and Ref. 68, p. 137 and Sec. 13 of Chap. VIII.

Ralph

 

Baierlein

, (, , ), Chap. 12.

L. D.

 

Landau

and

E. M.

 

Lifshitz

, , 3rd ed. (, , ), pp. –; and Ref. 66, Secs. 1, 16, and 44.

Richard W.

 

Hamming

, (, , ), pp. – and Secs. 8.12–8.13. See especially p. 3 and Sec. 8.13.

See, for example, Ref. 53, Chaps. 3 and 4.

Sheldon

Ross

, (, , ), 2nd ed., Sec. 1.6.

It is often stated that the exponential distribution is the only memoryless continuous probability distribution and that the geometric distribution is the only memoryless discrete probability distribution. However the geometric distribution can be expressed in exponential form, i.e., as a discrete exponential distribution. Aspects of this point are discussed in

Saeed

 

Ghahramani

, , 3rd ed. (, , ).

See Ref. 20, Complement $AIII$ (see especially Part 1) and the associated discussions in Ref. 19.

Donald H.

 

Kobe

, private communications ().

A probability of measure zero corresponds roughly to one chance in $∞$⁠, for example, the probability of obtaining exactly five given a uniform probability density over the real numbers from zero to ten. A probability of strictly zero corresponds to zero chance in $∞$⁠, for example, the probability of obtaining exactly 15 given the same, or the probability of 100 cm annual precipitation at a weather station whose annual precipitation is 50 cm in half of the years and 150 cm in the other half—the average of 100 cm never occurs in any year. For helpful discussions see

John E.

Freund

, (, , ), Chap. 22;

, edited by

W.

Gellert

,

S.

Gottwald

,

M.

Hellwich

,

H.

Kästner

, and

H.

Küstner

(, , ), 2nd ed., Chap. 14;

Isaac

Asimov

, (, , ), Chap. 5;

and

John D.

 

Barrow

, (, , ), especially Chap. 4 and pp. –.

See

Martinus

 

Veltman

, (, , ), Chap. 4 and 9.

Roger

 

Penrose

, (, , ), Sec. 26.8.

However, see Ref. 61, p. 252.

Reference 67, Sec. 10.6 (especially p. 306).

J. J.

 

Sakurai

, , revised ed. (, , ), pp. – and Secs. 5.6 and 5.8.

L. D.

 

Landau

and

E. M.

 

Lifshitz

, , 2nd revised ed. (, , ), Sec. 44.

E. M.

 

Henley

and

A.

 

Garcia

, , 3rd ed. (, , ), Sec. 5.8, pp. –, and 3.10, and Problem 10.31 on p. .

A.

,

Messiah

, (, , ).

W.

 

Rindler

, , 2nd ed. (, , ), Sec. 6.2–6.3 (especially Sec. 6.3). See also the reference cited in Sec. 6.2. In 1905, Einstein could prove only “that energy contributes to mass, without (necessarily) causing all of it”—it was still logically possible to assume that particles had inviolable “core” masses. “To equate all mass with energy (in 1905, and even in the subsequent early years of special relativity) required an act of aesthetic faith, very characteristic of Einstein.” Interpretations differ concerning what qualifies as the first complete derivation of $E=mc2$ and the date of its achievement—compare and contrast Refs. 71 and 72.

W.

 

Rindler

, , 2nd ed. (, , ), Secs. 26–27 (especially Sec. 27, wherein the two quotations cited from Sec. 6.3 of Ref. 69 are also given).

C.

 

Lanczos

, (, , ), pp. xi, 85–88, 96–105, 112, 139, 142–143, and 149–153. See especially pp. 85–86, 96–97, and 149–150.

H. C.

 

Ohnian

, (, , ), Chap. 7, and Notes and references for Chap. 7 on pp. –.

H.

 

Margenau

,

W. W.

 

Watson

, and

C. G.

 

Montgomery

, (, , ), Sec. 50.2.