If energy is not preserved, then Planck's constant is no longer a constant: a theorem Текст научной статьи по специальности «Физика»

UDC 530.145:51-73

IF ENERGY IS NOT PRESERVED, THEN PLANCK’S CONSTANT IS NO LONGER A CONSTANT: A THEOREM

V. Kreinovich, A. Ortiz

For any physical theory, to experimentally check its validity, we need to formulate an alternative theory and check whether the experimental results are consistent with the original theory or with an alternative theory. In particular, to check whether energy is preserved, it is necessary to formulate an alternative theory in which energy is not preserved. Formulating such a theory is not an easy task in quantum physics, where the usual Schroedinger equation implicitly assumes the existence of an energy (Hamiltonian) operator whose value is preserved. In this paper, we show that the only way to get a consistent quantum theory with energy non-conservation is to use Heisenberg representation in which operators representing physical quantities change in time. We prove that in this representation, energy is preserved if and only if Planck’s constant remains a constant. Thus, an appropriate quantum analogue of a theory with non-preserved energy is a theory in which Planck’s constant can change - i.e., is no longer a constant, but a new field.

1. Formulation of the Problem

Every physical law needs to be experimentally tested. Physics is a rapidly changing science, new discoveries are being made all the time, experimental discoveries that are often inconsistent with the existing physics and which lead to a development of new physical theories. Testing the existing physical theories is one of the main ways how physics evolves.

How physical laws can be experimentally tested. To test a physical law, we must:

• formulate an alternative theory in which this law is not valid (while others are valid),

• find a testable experimental situation in which the predictions of this alternative theory differ from the predictions of the original theory, and then

• experimentally check which of the two theories is correct.

Copyright © 2012 V. Kreinovich, A. Ortiz

University of Texas at El Paso (USA)

E-mail: vladik0utep.edu, aortizl90miners.utep.edu

Example. This is how the General Relativity theory (alternative at that time) was experimentally tested: by experimentally comparing the predictions of Newton’s gravitation theory - the prevalent theory of that time - with the predictions of the alternative theory; see, e.g., [3].

How can we test energy conservation law? A problem. One of the

fundamental physical laws is the energy conservation law. At first glance, checking this law is easy: even on the level of Newton’s physics, with the usual equations of motion

d2Xi =1 f 7,0 ' Ji, V U

at2 m

d2 Xi

relating acceleration ——i with fee force fi, there are many non-potential force

at2

fields fi(x) in which energy is not preserved.

The problem appears when we take quantum effects into account, i.e., when we consider the quantum equations. In quantum physics, the main equation -originally formulated by Schroedinger - has the form (see, e.g., [1])

i ■ h ■ dt = Hi, (2)

where i =f \f—1, i(x) is the wave function describing the quantum state, and H

non-potential force field, there is no well-defined notion of a total energy and thus, it is not possible to write down the corresponding quantum equation.

Discussion. The need to test the energy conservation law on quantum level is not purely theoretical:

• on a pragmatic level, serious physicists considered the possibility of microviolations of energy conservation starting from the 1920s [1];

• on a more foundational level, the intuitive ideas of free will seem to lead to possible energy non-conservation [2].

What we do in this paper. In this paper, we show how to form a quantum theory in which energy is not conserved. Specifically, we show that for quantum theories, energy non-conservation is equivalent to changing Planck’s constant. Thus, in quantum physics, checking whether energy is conserved is equivalent to checking whether Planck’s constant changes.

2. Analysis of the Problem

Schroedinger and Heisenberg representations of quantum physics: reminder. In quantum physics (see, e.g., [1]), states are described by elements of

a Hilbert space - e.g., of the space of all square-integrable functions i(x) - and physical quantities are described by linear operators in this space.

Historically, quantum physics started with a description by Heisenberg, in which states are fixed but operators change. Very soon, it turned out that in most cases, an alternative representation is more computationally advantageous -a representation in which operators are fixed but states change. This representation was originally proposed by E. Schroedinger and is therefore known as the Schroedinger representation.

Since we cannot use Schroedinger’s representation, we will use the Heisenberg one. As we have mentioned, the Schroedinger’s equation implicitly assumes the existence of (preserved) energy. Thus, to describe situations in which energy is not preserved, it is reasonable to use the Heisenberg representation.

Heisenberg representation: first approximation. In the Heisenberg representation, physical quantities like coordinates xi and components pi == m ■ of

the momentum vector are represented by operators. In the first approximation, the usual quantum mechanics is described by the usual Newton’s equations

dxi 1 dpi

~tt = ■ p^ -TT = 7^ (3)

dt m dt

xi pi

This description was first found by P. Ehrenfest (see, e.g., [1]).

The difference between the scalars and operators is that operators, in general, do not commute, i.e., in general, for two operators a and b, we have [a, b] = ab-ba = 0. Specifically, in the usual quantum physics, operators xi and xj corresponding to different coordinates commute with each other, operators pi and pj commute with

xi pi

[xi, xj] = 0, [pi,pj] = 0, [pi, xj] = i ■ h ■ 5ij, (4)

where the Kronecker’s delta is equal to 1 when i = j and to 0 otherwise. For the usual energy-preserving quantum mechanics, these commuting relations get

xi pi

equations (3).

[a, b]

the Poisson brackets (see, e.g., [1]). Namely, for arbitrary functions a(x,p) and b(x,p) of coordinates x = (xi,x2,x2) and momentum p = (p1,p2,p3), we have

[a, b] = i ■ h ■ {a, b} + o(h), (5)

where

{a b} d=f (

dpk dxk dxk dpk/

r de^ f da db da db \

{a-b} = L 5^ ' 7>x - Tix ' Tip . (6)

k

As an example, let us show what happens for the Heisenberg commutator [a, b] for

dpi

which a = pi and b = Xj Sinee a = prepends only on pi, we have

dpi dx ■

and = 0 Similarly, si nee b = Xj depends only on Xj we have 3

dpk

8.

ik

dxk

dXj - 8jk. Thus,

dpk

0 and

dx

k

When the force comes from a potential field, Planck’s constant is preserved. Let us show that in the potential field with potential energy V(x), when dV

fi = — -—, Planck’s constant is preserved. Indeed, let us assume that the com-

dxi

muting relations (4) hold at a certain moment of time t0. In particular, this means that [pi, xj] = i ■ h ■ 8ij. Let us show that - at least in the first approximation - this

d

relation is preserved, in the sense that — [pi,xj] = 0.

[a, b] = ab — ba

d d da db db da

— (la, bl) = — (ab — ba) = — b + a—-------------------------------i~a — b—

dt dt — dt dt — dt — dt

da

~r,b

dt

+

db

a,dt

Thus, we have

d

^(|pi,xj ]) =

dpi

dt

, xj

+

pi

dxj

dt

Due to Ehrenfest equations, we have = fi and ^ = — ■ pi, so we have M dt J dt m F

d 1

dtt ([pi,xj]) = [fi,xj] + m ■ [pi,pj].

(7)

fi

have [fi,xj] = 0. Since all the components of the momentum commute, we have

d

[pi,pj] = 0 Thus, we conclude that —([pi,xj]) = 0.

Similarly, we can conclude that the second derivative of the Heisenberg commutator is also equal to 0. Indeed, by differentiating both sides of the equation (7), we conclude that

% ([pi’xj])

dt

, xj

+ — ■ [fi,pj] + — ■ [fi,pj] + — ■ [p^ fj]. (8)

mmm

Here, since fi depends only on coordinates, we have

dfi dxe 1

dt

E

dxo dt

so

dfi

d2 1

* (fe-xj') = m

Egxt ■ pc'xj

t

+ 2[fi,pj] + [pi, fj]

Thus, to prove that this second derivative is equal to 0, it is sufficient to prove that the expression in parentheses is equal to 0. In the first approximation, this expression is proportional to the sum S of the corresponding Poisson brackets

S = {f ■ pt,xj \ + 2{fi,pj} + {p^ fj};

S

xj xj

dfi I = ^ f 8 8 = f

L dx, ■ pt,xjj = EE dx, ■8kt ■8jk = dXj ■

S pj

get

u \ dfi 8 U

{fi,pj} = — 2_ dX7 ■ j

dxk dxj

k

Similarly,

Thus, we have

dfj dfi

S =^j -

dxi dxj

dV

For the potential field, we have fi = —77— and therefore,

dxi

dfi d2V

Hence, we have

dxj dxidxj

S = —_dH_ + = 0.

dxidxj dxidxj

What happens when energy is not preserved: an example. We are interested in situations where energy is not preserved. Let us start our analysis with the simplest such situation of the friction force fi = —k ■ vi, i.e., force of the type

fi = — k0 ■ pi, where k0 == —. In this case, from the formula

m

dt (\Pi,xj i)=|/i>xj i+m

by using [pi,pj] = 0, we get

d

dt ([pi,Xj]) = —k0 ^ [pi,Xj].

In other words, for h == [pi,xi], we have a differential equation

dh

— = —ko ■ h.. dt

dh

From this equation, we conclude that — = —k0 ■ dt hence ln(h) = const — k0 ■ t,

h

and h(t) = const ■ exp(—k0 ■ t). We know that h(t0) = i ■ h, hence

h(t) = h(t0) ■ exp(—k0 ■ (t — t0)).

At the initial moment t0, we have h(t0) = i ■ h. So, the above equation means, in ect, that Planck’s c decreases with time.

effect, that Planck’s constant h == [pi,,Xi] is no longer a constant - it exponentially

Discussion. Let us show that the same phenomenon - of Planck’s constant no longer being a constant - occurs for every theory in which energy is not preserved.

3. Main Result

Formulation of the main result. Let us consider the general case, when fi x p

will show that if in the quantum version of this theory, Planck’s constant remains a constant, i.e., we have [pi,Xj] = i ■ h ■ 8ij for all moments of time, then the field

dV

fi is a potential field, i.e., has the form fi = — -— for some function V(x).

fi

a constant.

Proof. If Planck’s constant is a constant, this means, in particular, that we have

d

— ([pi,Xj]) = 0. Explicitly differentiating the left-hand side, we conclude that

[fi,Xj] + — ■ [pi,pj] = 0. Since [pi,pj] = 0, we get [fi,Xj] = 0. In the first

approximation, this means that the corresponding Poisson bracket is equal to 0: {fi,Xj} = 0. Since Xj depends only on the coordinate, we get

/ f i = dJi 8 = dJi = 0 {fi,Xj} = 1. dpk ■8kj = dpj =°-

fi pj fi

fi

fi

d2

time derivative — ([pi,Xj]), we can repeat arguments from the previous section and conclude that in the first approximation, this second derivative is proportional

t0 S =2. dJl-2. dfi

dxj dxj

So, from the fact that the second derivative is equal to 0, we conclude that S = 0,

i.e., that

J = J

dxj dxj

for all i and j. It is known that these equalities are necessary and sufficient

dV

conditions for the existence of a field V for which fi = — -—. Thus, we have

dxi

proved that ^ is indeed a potential field. ■

Discussion. Theories in which Planck’s constant is no longer a constant but a new physical field s(x) have been proposed; see, e.g., [4].

It should be mentioned that we may need to go beyond proposed theory: indeed, these theories only consider a scalar field s(x) corresponding to

[pi,Xj] = i ■ h ■ s(x) ■ 5ij,

while, in general, the commutator [pi,Xj] can be an arbitrary tensor.

Acknowledgments

This research was partly supported by the National Science Foundation grants HRD-0734825 and HRD-1242122 (Cyber-ShARE Center of Excellence) and DUE-0926721, by Grant 1 T36 GM078000-01 from the National Institutes of Health, and by a grant on F-transforms from the Office of Naval Research.

One of the authors (VK) is greatly thankful to his late father Yakov R. Kreinovich for inspiring ideas and discussions about energy conservations.

References

1. Feynman R., Leighton R., Sands M. The Feynman Lectures on Physics. Boston, Massachusetts : Addison Wesley, 2005.

2. Kreinovich V. In quantum physics, free will leads to nonconservation of energy // Journal of Uncertain Systems. 2013. V. 7, to appear.

3. Misner C.W., Thorne K.S., Wheeler J.A. Gravitation. New York : W. H. Freeman, 1973.

4. Phipps T.E. Jr. // Found. Phys. 1973. V. 3. P. 435.