On the measurement of the zero point energy
Posted on Sunday, February 20, 2005 @ 23:21:37 GMT by vlad
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In Sarfatti_Physics_Seminars Kay zum Felde writes: On the measurement of the zero point energy
1.Introduction
Quantum mechanics is based on the uncertainity principle. This says, that the energy of a quantum mechanical system cannot be specified exactly within a time period. The same is the case, if we want to measure momentum and the corresponding position of a particle at the same time. According to quantum mechanics every particle can be regarded as a wave and its frequency is associated with its energy. The frequency is multiplied with the constant hbar discovered by Planck [1]. This constant indicates, that every quantum particle has the same origin. Particles differ from each other by their quantum properties, like their energy, momentum, spin, charge etc.
If we remove all matter from space, we are left with what is called vacuum. Quantum mechanics predicts, by the uncertainity relation, that the vacuum is not empty. Virtual pairs of particles can be created and destroyed. These processes are going on and on and they leave a rest energy. We cannot measure these particles, but we can measure the rest energy they leave. This is called zero point energy.
It can be shown, that the
zero point energy is needed to fulfil the principle of least action in the case of the
semi-classical harmonic oscillator [2]. A physical principle is, that energy corresponds
with a positive value. This principle is the basis to 'introduce' the zero point energy.
The solution to the quantum mechanical harmonic oscillator problem needs the zero
point energy, otherwise there could exist particles with negative energies [3].
Haisch, Rueda and coworkers have shown, that the zero point energy can be regarded
as inertia mass [4,5,6,7,8]. They claim in their semi-classical approach [5], that
inertia could have an extrinsic origin. Since Newton's Principia, it is the usual
assumption for many physicists to regard inertia as an intrinsic 'property' of mass [7].
The main interest of Haisch, Rueda and coworkers is the origin of inertia. Our interest
is to find an effect or a force, which is driving, which is the origin of the zero point
energy. This does not reflect, whether the origin of inertia is the zero point energy or
not; it proposes an ansatz for the problem, where the high or infinite energy
fluctuations of the vacuum are resulting from, not what they are.
This problem seems to reach back to the origin of the universe. Stephen Hawking is
explaining in his book "A brief History of Time" the possible process of the beginning
of our universe. After the big bang, the temperature was relaxing to about one billion
Kelvin. This is the temperature inside stars. The protons and neutrons were combined
to kernels and the kernels to Deuterium and later to Helium. Soon, some hours after
the big bang, the creation of Helium and other elements should have been
completed. The universe was thereafter inflating without any remarkable events.
In a lecture, Hawking and Penrose [9] discuss the problem of null-geodesics. This is
an effect dedicated to time-like or space-like solutions of the Einstein Equations. It
leads to naked singularities, which is basically the same as a null-geodesic. Null-
geodesics are trajectories in spacetime, which are past-incomplete. This means, the
future of the solution of the Einstein equations doesn't show a past end point. A
solution, which doesn't show any naked singularity would be not time-like nor space-
like, thus a null-geodesic and light-like.
It is also possible, to find plausible theories, which are showing, that virtual particles
are able to become real particles. This was a discovery by Hawking [9]. Black holes are
believed to 'emit' and absorb particles. These particles which are disappearing in the
black hole are different from these which are 'emitted'. Only their energy is the same.
These particles are coming from virtual pairs of particles. These virtual pairs of
particles are 'created' within the vacuum near the event horizon of the black hole [10].
Now, near the event horizon of a block hole, it can happen, that one particle of the
virtual pair of an electron and a positron, for instance, can be absorbed by the the
black hole [10]. This is due to the enormous gravitation field of the black hole. The
particle can have negative energy, fall into the black hole and become a particle or
anti-particle. If we are far away from the black hole, we believe, the black hole was
emitting the partner of the absorbed, now real, particle, which could be able to cross
the event horizon.
The question is basically, how can Quantum Mechanics be combined with Classical
Physics to explain phenomenons like quantum fluctuations and the big bang. We start
this conception by explaining our main point, which is restricting the spectrum of
virtual particles in quantum fluctuations. This restriction is in our view based on the
measurement of time and distance in Relativity. This is explained in section 2. In
section 3, we give brief overview on the recent development of the theory on the zero
point energy. Section 4 is discussing the universe as a gigantic harmonic oscillator
and the form of the origin momentum, which could be initiated our universe. Section
5 is dedicated to the quantum fluctuations.
2. Measurement in Classical and in Quantum Physics
I plan to compute a simple example, starting from the point of view, that the universe
was beginning as an oscillating 'device'. By that idea, we mean, that the density of
mass was oscillating with a special frequencies or as a superposition of harmonic
frequencies. Later it turns out, that this is an interesting approach. Our idea was born
by an intuitive approach starting with the quantum mechanical principle, that every
particle is represented by a frequency. In classical physics, a harmonic oscillator is
build by applying a periodic external force to the system.
Even in classical physics we can only observe frequencies. To
examine this, we come back to Einstein's question of {it simultaneous} events
[11]. Einstein defines a syncronized event by defining time. 'Time' is therefore
synchronized, if two clocks are synchronized. The procedure to achieve synchronized
clocks is based on the measurement of time intervals [11]. Two events can in these
terms only be simultaneous, when they are happening at the same place. We compute
the distance between two points in spacetime. To simplify this, we consider two
bodies in flat spacetime, which are not moving with respect to each other. The
distance
egin{equation}
s=sqrt{sum_{mu=1}^{4}Delta x^{2}_{mu}}=sqrt{sum_{mu=1}^{4}(x_{mu,1} -
x_{mu,2})^2}
end{equation}
is measured by sending a ray of light from $vec{x}_{1}$ to $vec{x}_{2}$ and
measuring the time the ray of light needs to reach the second body and is coming
back. This is twice the light needs to run from $vec{x}_{1}$ to $vec{x}_{2}$. The
time frame of observing is therefore an intrinsic property of Special and General
Relativity. Can we measure the distance through any other tool or procedure in
Relativity, which is not based on the observation of a ray light moving from point $A$
to $B$ and back ? No. What, if the clocks are synchronized and we are dealing with
the most simple special case, presented above. In practice, we don't know whether
the second event is moving, we have to synchronize the clocks to be sure, that we
deal with the same time. This means, the assumption we made, that the clocks are
synchronized is going to far. We need to synchronize the clocks. On the other hand, if
we are aware of velocity of light in the given medium, we can use the procedure
which sends one ray of light from event $A$ to event $B$ and back.
This means, even in classical theories, we too are dealing with time frames. The finite
velocity of the light and the fundamental idea, that two bodies have a minimum
distance between themselves, leads to a restriction of the spectras of the frequencies
in all theories. Quantum gravity propagates the so-called Planck length [12]. This is
the nearest distance, that two particles carrying a mass can reach. Its value is
egin{equation}
r_{pl}simeq 1.62x10^{-35} m
end{equation}
Mathematically we can bring the two particles so close together, that their distance is
zero, but this lacks the physical appearence of the gravitational field. this is becoming
infinite at $r=0$. Thus the particles are physically forbidden to come together at zero
distance. At the Planck distance, a particle called graviton is believed to be exchanged
between the two massive particles [12]. The graviton is the mediator between masses.
ewline
Does the observation and the theory, that we have to consider time frames
automatically require oscillations in time in any measurement ? A frequency is a rate
of a repition of a periodic disturbance. As we pointed out earlier, we measure the
distance between two points by measuring the time, which the ray of light needs to
travel from one body to the other and back. This is a full time period. A full time
period is always associated with a frequency. The external disurbance by the ray of
light, we mentioned, was simply consisting of a rate of one. The measurement of time
is made by clocks. These clocks follow intrinsic systems, which are using frequencies.
Einstein points out, why all distance measurements are made by means of clocks
cite{ein}.
ewline
The distance of stars, of course cannot measured by such a simple system. they are
measured by observing their relative velocity. And this is done by means of the
Doppler effect. The spectras of stars are compared with the light spectrum in the
laboratory on earth. These spectras aren't influenced by any movement [13]. Thus we
obtain frequencies.
2. The zero Point Energy
The quantum mechanical harmonic oscillator needs to consist of a zero point energy,
which follows from the principle that energy is always positive. The energy of a
particle can be measured by observing its frequency. hbar is positive, thus the
frequency must be positive. It follows, that it is not possible to travel back in time, if
we apply the principle of positive energy to Special and General Relativity [1]. We
observe always frequencies in Quantum Mechanics and also in Special and General
Relativity.
With their work, Rueda, Haisch and coworkers [4,5,6,7,8] have established an idea of
inertia having an extrinsic orgin. Their hypothesis is the arising of inertial reaction
forces from the interaction of material particles with local vacuum fluctuations [6]. In
[8] Haisch, Rueda and Puthoff stated that the zero point energy is a quantum
mechanical effect. As earlier pointed out, the origin of the zero point energy is that
the energy of any particle is always positive . So the zero point energy arises from
quantum mechanical calculations, but the origin of the zero point energy is that the
energy is always positive. A true quantum mechanical effect of the solution to the
quantum mechanical harmonic oscillator is, that the energies are discontinuous. They
present a formula for a uniform, linearly accelerated black body radiation field. This
formula is based on the Unruh effect, which says that an "observer who accelerates in
the conventional quantum vacuum of Minowski space will perceive a bath of
radiation, while an inertial obeserver perceives nothing. In the case of linear
acceleration, for which there exists extensive literature, the response of a model
particle detector mimics the effect of its being immersed in a bath of thermal
radiation" [14]. This heath bath is quantum phenomenon [8]. The formula
egin{equation}
ho(
u)=frac{8pi
u^2}{c^3}frac{h
u}{2}d
u
end{equation}
becomes through the acceleration process
egin{equation}
ho(
u, T_{a})=frac{8pi
u^2}{c^3}Biglbrack 1+Big(frac{a}{2pi
c
u}Big)Big
brackBiglbrackfrac{h
u}{2}+frac{h
u}{exp(frac{h
u}{kT_{a}}-
1}Big
brack
end{equation}
Haisch et al. [8] follow, that the zero point mass is given by
egin{equation}
m_{zp}=Big(frac{V_{0}}{c^3}Big)int eta(omega)frac{hbaromega^3}{2pi^3
c^3}=frac{V_{0}}{c^2}int eta(omega)
ho_{zp}domega
end{equation}
Haisch et al state [8]:
"We are suggesting, that the mass of a particle is determined by a resonance
frequency omega_{0} and that the mass of a composite entity can be radically
different from the sum of the indivdual masses, because of the changes in the
resonance to binding forces."
This is practically bringing together the classical concept and the quantum
mechanical concept of energy. Mass equals frequency equals energy up to a constant
factor. This is a remarkable result, but what does it help us ? The problem is still:
where does the zero point energy is really coming from ? What physically force is
leading to it ? The eigenvalue of the zero point energy is frac{hbaromega}.
What happens if the zero point energy doesn't exist ? That would mean, negative
values of the energy are possible. Dirac's infinite sea of electrons wouldn't be infinite
anymore. There were possible energies were the electron from above the gap, would
fall into. If this sea is finite, these not occupied energies would be occupied after
some time (because the particles will search for lowest energy states), and the
negative energies would not be free anymore. The restriction, that the number of
electrons in Dirac's sea is finite would lead to something, solid state physicists are
calling Fermi energy [15]. The vacuum would be the Fermi energy. The Fermi energy
is the point of energy, where the energy change is measured by putting an additional
particle into the system. At the Fermi energy, this change in energy equals the total
internal energy in this thermodynamical system. The Fermi energy would be on the
level of the vacuum.
ewline
In the case of Quantumelectrodynamics, the zero point energy describes the energies
of virtual electron-positron pairs (see for instance [16]). frac{1}{2}hbaromega is an
expectation value; it is therefore a mean value. This seems to give respect to the
process the virtual pairs are taking, in which they are crossing the Dirac gap and are
falling back. This is in average a process where one virtual photon is absorbed and a
virtual electron-positron pair is produced. This indicates that in average half a photon
is needed to produce two particles, the electron and the positron.
ewline
If we assume for now, that the Dirac sea is finite, that the vacuum energy is behaving
like the Fermi energy in a solid. The zero point energy would be fixed to the Fermi
energy of the universe.
The idea presented now is intuitively coming from the following facts and
suggestions:
(1) We can only observe frequencies. This is due to an intrinsic feature of Special and
General Relativity. Basically the main reason is the finite velocity of a ray of light c.
The measurement of a distance needs the use of a ray of light, which travels from one
point in space vec{x}_{1} to the second one vec{x}_{2} and back. This menas, that in
our reality, that what we observe, we don't have any indication, that space is
continuous.
ewline
(2) The quantum mechanical harmonic oscillator consists of a zero point energy, if we
demand the energies to be positive. The eigenvalues of this system are
discontinuous.
ewline
(3) The idea, that the zero point energy is build of inertial masses, which are
consisting of a resonance frequency. This statement is coming from a semi-classical
approach and shows the internal relation between mass and frequency.
ewline
(4) The Planck length is the nearest distance two mass particles can reach.
ewline
(5) If the spectrum is finite, what the Planck length implies, and what also the finite
light velocity implies, then we assume, that the vacuum is something like a
thermodynamical Fermi energy, which is, in our case, the change of the internal
energy of the universe by changing the number of particles.
All these ideas are leading to the conclusion, that inertial mass is oscillating and that
the number of particles is finite in the universe. This would mean, that the Dirac sea
is not infinite, because it is build up on electrons. We cannot put infinite electron-
positron pairs out of the Dirac sea. This is not possible, because two electrons
cannot, classical, be at the same place. Because their distance is limited, we measure
a frequency, which is finite.
I`m aware, that this last statement is a conclusion based on quantum mechanical and
classical concepts. I said, that the frequency of the quantum mechanical system is
constrained, because the distance between these two particles is restricted,
classically. Position and momentum don't have the same meaning in Quantum
Mechanics and Classical Physics. In Quantum Mechanics, they appear as differential
operators. In momentum representation, the position operator is characterized by a
change in momentum and momentum is simply a number. This reflects the
uncertainity relation. I`m thus aware: the classical idea, that particles cannot be at the
same time in the same place is making the approach a semi-classical theory.
References:
[1] M. Planck, Verh. dt. phys. Gesellschaft 2 (1900) 237; Ann. Phys. 4 (1901) 553
[2] D. Bohm, Quantum Theory, reprint, Dover, New York (1989), p. 282
[3] D. Bohm, Quantum Theory, reprint, Dover, New York (1989), p. 299
[4] A. Rueda, B. Haisch, Found. Physics 28 (1998) 1057
[5] A. Rueda, B. Haisch, Phys. Lett A240 (1998) 115
[6] Y. Dobyns, A. Rueda, B. Haisch, Found. Phys. 30 (2000) 59
[7] A. Rueda, B. Haisch, R. Tung, arXiv:gr-qc/0108026v1 (2001)
[8] B. Haisch, A. Rueda, H.E. Puthoff, arXiv:physics/9807023v2 (1998)
[9]S.W. Hawking, R. Penrose, The Nature of Space and Time, Lecture presented at the
Isaac Newton Institute, Cambridge,
[10] S.W. Hawking, Das Universum in der Nussschale (Rowohlt 2003)
[11] A. Einstein, Ann. Phys. 17 (1905) 891
[12] J. Polchinski, String Theory, Vol I, 4th ed., Cambridge University Press,
[13] M.V. Berry, Kosmologie und Gravitation, 1st ed., Teubner, Stuttgart (1990), p. 8
[14] P.C.W. Davies, K. Dray, C.A. Manogue, Phys. Rev. D53 (1996) 4382
[15] O. Madelung, Introduction to Solid-State Theory, 3rd ed., Springer, New York
(1996), p. 26
[16]R.P. Feynman, Quantenelektrodynamik, 4th ed., Oldenbourg, Munich (1997)
Kay zum Felde
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