Let us assume that the string can move up and down. The classical hamiltonian contains energy from the stretching of the string:
1/2 (δ_x φ)^2,
as well as from the vertical movement kinetic energy of the string:
1/2 π^2.
We assume that there is a lowest energy state |0 >. Energy is added to the string by adding standing waves which have some number n of nodes.
In quantum mechanics, the string cannot stand still even in the lowest energy state. It moves up and down.
We do not know if there exists a lowest energy state if we couple the field to a detector
According to the Wikipedia article, one can prove the existence of a lowest energy state, the "vacuum state", only in a few simple cases through constructive quantum field theory.
The interacting system of a detector and the field (the string) is complex. It is probable that we cannot prove the existence of a lowest energy state.
Every vibration mode of the string contains energy, 1/2 h f, where f is the frequency. The total energy is infinite. The detector disturbs the system, and the disturbance might free an infinite amount of energy from the system.
The infinite energy of the string is normally canceled by "renormalizing" the energy of the lowest energy vibration state to zero. But that does not help if we couple a detector to the field, since the interaction might create an even lower energy state.
A particle model solves the problem of infinite energy in quantum field theory
The problem of infinite energy in a quantum field can be solved by moving to a particle model. The infinite number of vibration modes in "empty" space do not exist. Only particles exist. Their wave phenomena is caused by a path integral.
In a particle model, empty space is truly empty. There are no "zero point fluctuations" of various fields in empty space.
A crystal of atoms in the lowest energy state and a moving detector: friction
Suppose that we have a crystal of atoms in the lowest energy state. Then we move a detector close to them, or even through the electron orbits of the atoms. The detector may disturb the atoms. Kinetic energy of the detector can be converted to heat, which may excite the atoms as well as the detector itself.
There is kind of friction between the atoms and the detector. Since atoms have mass-energy, they can absorb momentum from the moving detector, and free kinetic energy as heat.
If we assume that empty space is full of oscillators in their lowest energy state, then moving a detector at a constant speed could disturb the system. But empty space has no mass-energy (since we renormalized the infinite energy away). It cannot absorb momentum, and consequently, friction is prohibited.
Another way to say this is that Lorentz covariance prohibits such behavior, since we assume that in empty space all inertial frames are equivalent. There cannot be friction in one frame and not in another.
However, the interaction with the detector might allow the space to fall into a lower energy state. Then even a static detector might see some kind of "Unruh radiation" coming from empty space.
Also, renormalizing away the infinite energy of empty space is a dubious procedure. If we have a wildly oscillating string and let our fingers slide on it, we expect to feel violent oscillation. We cannot remove the phenomenon simply by deciding in our mind that the string has zero mass-energy and our fingers cannot feel friction because then Lorentz covariance would be violated.
Derivation of Unruh radiation
Luis C. B. Crispino et al. (2007) present a short derivation of Unruh radiation.
Let us have a detector on an accelerating rocket. It is sensitive to a frequency f of scalar massless Klein-Gordon field waves.
Does the detector see a frequency f wave? The transformation of an imagined frequency f wave to an inertial frame wave is a chirp. The inner Klein-Gordon product of the chirp with the vacuum state |0 > of the field is not zero. The negative frequencies in the transformed wave have a non-zero inner product with the vacuum.
This is because we can write a positive frequency mode f_i in the accelerating frame as a Fourier decomposition
f_i = Σ α_Ii* f_I - β_Ii f_I*,
I
where f_I is a positive frequency mode in the inertial frame, and f_I* is a negative frequency mode in the inertial frame. The algebra then tells us that the annihilation operator
â_i = Σ α_Ii â_I + β_Ii* â_I-dagger.
I
That is, the annihilation operator in the accelerating frame is partly an annihilation operator in the inertial frame, but partly a creation operator (the dagger version).
Usually, the β are extremely small. Only if the acceleration is > 10^25 m/s^2 they would become significant.
The number operator of the accelerating frame, when applied to the vacuum of the inertial frame, gives a non-zero value. This is because the annihilation operator above is partly a creation operator in the inertial frame. As if there would be particles in the accelerating frame.
Negative frequencies are solutions of the Klein-Gordon equation, too. But the creation operator only creates positive frequencies. Where did the negative frequencies disappear? Apparently, they were declared "unphysical" and discarded. The negative frequency solutions may be understood as solutions of negative energy.
The above canonical quantization does not make sense in the classical limit
A detector observes a physical field. It could be a classical real scalar Klein-Gordon field.
The quantization in the previous section does not handle that case. We would need the negative frequencies, so that we can sum a mode
exp(-i (E t - p x))
and its complex conjugate and get a real value.
The previous section does not have a sensible classical limit. It maps a classical positive frequency wave in the accelerating frame partly to an "unphysical" negative frequency wave in the inertial frame. In classical physics that cannot happen.
It looks like that the textbook example of canonical quantization is fatally flawed. All fields occurring in nature must admit negative frequencies, because those frequencies inevitably appear in accelerating motion. There cannot exist a field in nature where only positive frequencies are allowed.
The electromagnetic field, for example, definitely allows negative frequencies. We discussed them in the previous blog posting. A right-handed photon becomes partly left-handed if observed from an accelerating rocket.
The Unruh effect might be observation of the zero-point fluctuation of a field?
The derivation of Unruh radiation seems to be fatally flawed. But the phenomenon might still exist if our detector can detect energy in the hypothetical zero-point fluctuations of the assumed quantum fields filling the vacuum.
Actually, it is probable that a detector, accelerated or not, can observe such energy.
detector
●
/ rubber
/ band
-----------------------------------
string in the lowest energy state
(lowest without interaction)
Suppose that we have a string in its lowest energy state. It still vibrates, according to standard quantum mechanics.
We bring a detector close to it and attach the detector via a rubber band to the string. The rubber band represents an interaction. It would be surprising if there would be no response whatsoever in the detector. The system is no longer the same when the interaction is present.
Hypothetical Unruh radiation is just the tip of an iceberg of the strange phenomena which are implied by zero-point fluctuations.
Another one is the hypothetical infinite energy of the vacuum.
If we believe in zero-point fluctuations of the electromagnetic field, then a single electron interacts with infinite energy in those fluctuations. Why the electron sits still if it is bombarded by infinite energy?
Also, a single electron couples the modes, containing infinite energy, all together. How does the infinite energy react?
It looks like we have to abandon the concept of quantum fields filling the entire space. Classical fields are ok because they do not have zero-point energy. A particle model with a path integral might be a way to avoid the most serious problems of infinite energy.
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