**Absurdities in Modern Physics:** **
A Solution**

Paul Marmet

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Quantum Mechanics uses a relation that is interpreted as being a fundamental limit of resolution in physics. It is called the uncertainty relationship. This uncertainty principle is so consequential in physics that Cramer [3.1] states:

The uncertainty relationship can be written in different ways. Let us begin with the relationship as written by Heisenberg [3.2]:

3.1 |

DE = resolution in energy, Dt = resolution in time, h = Planck constant.

To avoid confusion, let us first state the argument given by Heisenberg [3.2], establishing a limit of resolution. The starting point is the one that characterizes any monochromatic wave packet of frequency n. A continuous wave is cut up into pieces by a time shutter, forming wave-packets of duration Dt. Using the Fourier transform, Heisenberg has observed that the spectrum of a wave packet, as given by a spectroscope, is limited in resolution. A wave packet limited by a time interval Dt shows a wider bandwidth DE when Dt is smaller, as given by Eq. 3.1. That equation, resulting from the use of the Planck relationship E = hn, led Heisenberg to the uncertainty principle.

Heisenberg hypothesized that the wave packet described represents one single photon. His mathematical analysis however is solely based on the description of a wave phenomenon. There is nothing in the wave packet model that implies that it is valid only for a unique small amount of energy. It appears useless to claim that a wave packet is the description of a single photon when the results are equally applied to any number of photons. If the Heisenberg uncertainty relationship characterizes only individual photons, we would expect to obtain a different resolution at higher intensity. We know that the uncertainty relationship is used to calculate the limit of resolution of telescopes and microscopes. However, the resolution of telescopes and microscopes is totally independent of the intensity of the signal observed. Even the gedanken "Heisenberg microscope" experiment uses the classical resolution in optics Dx = 2lsin(q) which is certainly independent of the intensity. Consequently, the Heisenberg's model is valid for any amplitude of the wave packet.

The mathematical transformations done by Heisenberg are certainly correct. Eq. 3.1 is a result that is a consequence of the Fourier transform of the wave packet. However, Heisenberg gives a physical interpretation of the result. He claims that the limit of resolution predicted by the mathematical transformation corresponds to a physical limit. We must therefore examine whether the limit imposed by Eq. 3.1 has physical implications or is simply a limit of resolution of one popular particular instrument, namely the spectroscope.

*3-2 Heisenberg's Wave
Packet
Description.*

The
physical
aspect of the problem is illustrated in the following way. In the case
of Heisenberg's model, we consider an initial wave with an unlimited
coherence
time. Within a wave packet, coherence is maintained at one frequency as
long as the wave maintains its phase relationship. The amplitude of the
signal varies according to a time-dependent sine function, as
illustrated
on Fig 3-A.

Heisenberg's
calculations are applied to the wave packet formed when the unlimited
sine
wave is cut up by a time shutter, between t_{1} and t_{2}
as seen in Fig. 3-A. Let us examine that wave packet in more detail.

Experimentally,
there are many ways to take measurements of a wave packet but at the
time
Heisenberg developed his model, only one method was used. It was
standard
spectroscopy. With the use of spectroscopy, one can easily obtain the
information
about the frequency and the bandwidth of the signal as illustrated on
Fig.
3-B. However, a spectroscope does not reveal all the existing
information
contained in a wave.

We know that different waveforms of wave packets can, when observed with a spectroscope, lead to exactly the same spectrum. This can be illustrated by a wave packet made from a wave with increasing amplitude or from a wave packet with decreasing amplitude. These two different wave packets can present spectra that are identical in frequency and bandwidth even if the initial wave packets have different shapes.

It is well known that a spectroscope is an instrument giving the total amplitude of the Fourier transform. The phase relationships between the frequency components are not measured by a spectroscope. Consequently, the spectroscope is losing experimentally the same amount of information as when neglecting the phase in the Fourier transform. The real as well as the imaginary parts of the Fourier transform are necessary to reconstitute the information about the phase. Therefore the spectroscope is a useful but limited method of studying wave packets, since much information is necessarily lost.

Detectors giving incomplete information are very common in physics. If we represent a signal by the description of its derivative, we clearly lose the integration constant. Another example occurs when we consider the energy given up to a wall that absorbs nitrogen molecules (from air). If we can make a precise measurement of the energy absorbed by the wall, following the absorption of a nitrogen molecule, we cannot deduce what the velocity of the nitrogen molecule was before it hit the wall. The reason is that we leave out of account what the rotational and vibrational energy of the molecules were before they hit the wall. The method of measurement of energy transmitted to the wall gives an incomplete account of the kinetic energy of molecules.

*3-3 Static Analysis.*

Let us
consider
the following example, illustrated on Fig 3-C. Let us generate a sine
wave
at frequency f_{1}, having amplitude A_{1},
during one cycle and a quarter (between 0 and 450 degrees).

This is
identical
to a continuous wave packet, with a cut such that only one and a
quarter
cycles are saved. Such a description is perfectly clear and complete,
and
anybody can reproduce such a wave packet from that description without
losing information.

In order to
study the wave packet, let us consider taking numerous samplings of the
amplitude at different times of the signal presented on Figure 3-C.
From
these data the original waveform can be faithfully reproduced.
Furthermore,
the exact original frequency can be calculated by fitting (mathematical
fitting by computer) the observed data between t_{1}
and t_{2} with a sine wave. That
mathematical
fitting must be able to take into account three parameters:
a)
the frequency, b) the phase and g)
the amplitude. The fitting is made on the original undisturbed wave and
not on the sharp cut induced by the experimenter. It is that sharp cut
that widens the spectral lines in spectroscopy.

The result of such a fitting leads to the exact frequency of the wave packet and the accuracy achieved can be as good as noise and technology will permit. The fitting will not give a wide bandwidth as it does in the case of the Fourier transform. Using the frequency, the phase and the amplitude obtained by the fitting, the wave packet can be reproduced without Heisenberg's uncertainty.

There is clearly nothing fundamental in physics about losing information when that loss of information is solely the consequence of the use of a mathematical transformation - the Fourier transform. Giving a fundamental meaning (like the uncertainty principle) to the limitation of the Fourier transform is like inventing another fundamental principle in physics to describe the information loss based on losing the integration constant when taking the derivative of a signal. There is no physics involved, only properties of mathematics. We will now study another method that, contrary to the use of the Fourier transform, will not lose the information (the phase) contained in the original wave packet.

**3-4 Characteristics of
Some
Phase-Sensitive Frequency Meters.**

Other than the static analysis mentioned in section 3-3, to explain basic principles, fast dynamic frequency meters using phase-sensitive detection have been developed in different frequency ranges. For example, an instrument called an automatic wave meter [3.3], uses phase-sensitive detection to measure accurate frequencies during short periods of integration. The instrument uses the fringe-counting Michelson interferometer technique and can count fringes while taking phases into account. The instrument typically counts electronically one million fringes in one second and can distinguish 1/50

Another method involves the use of phased-locked loops [3.4]. To describe the principle involved, we give here a description of the more fundamental phased-locked loops method to measure frequencies. Phased-locked loops circuits are commonly used in all color TVs, to adjust the phase controlling the color on every sweep of the electronic beam on the TV screen.

**3-5 Basic Principles
Involved
in Phase-Locked Detectors.**

The principle involved in phase-locked detectors corresponds to the following simplified procedure. Let us consider the wave packet described in Fig. 3-A. As soon as the electromagnetic signal enters the system at time t

In practice, there are a few cycles that are wasted during the initial period of electronic phase adjustment but this is negligible compared with the total of one million cycles usually counted. That knowledge of the number of cycles, with an accuracy of ±0.02 cycle, provides much more accurate information about the wave parameters than can be detected by a spectroscope, because the phase relationship is now known and the detector is not handicapped by a simple spectroscope. One of those instruments [3.3] could give reproducible results within ±0.02 cycle. This is smaller than 1/2p, contrary the Heisenberg uncertainty principle. It is smaller than 1/15p.

**3-6 Confusion between
Poor
Instrumentation and a Basic Phenomenon.**

Heisenberg was misled because he assumed that the only way to measure a wave packet was by measuring its spectrum. He even dared to claim that no other information than that obtained with a spectroscope can even exist. Heisenberg's interpretation is compatible with the Berkeley-Copenhagen interpretation on non-realism. According to the Berkeley-Copenhagen interpretation, the wave packet (or the photon) is created at the collapse of the wave function and does not exist independently of the observer. Heisenberg never tried to measure nor suggested measuring the frequency by using a more sophisticated method than the spectroscope.

The limit suggested by Heisenberg can now be interpreted either as an instrumental limit due to the particular use of a spectroscope or as a mathematical limit due to the information lost while using the Fourier transform without the imaginary part. In the case of a phase-sensitive detector, there is no fundamental limit. With a phase-sensitive detector, the limit of resolution depends only on the limit due to noise. It is unfortunate that Heisenberg could not distinguish between the loss of information given by a poor instrument and a fundamental principle.

**3-7 Hidden Criteria.**

We have seen that the Heisenberg uncertainty relationship is not fundamental in physics. We will now see that the amount of information lost, described as h/2p, is not absolute but quite arbitrary. It depends on the units chosen.

Let us go back to Heisenberg's model to describe the wave packet, as illustrated on Fig. 3-A. We will show how one can obtain three different limits of resolution, using a method compatible with the one used by Heisenberg. The results will be discussed afterwards.

From Fig. 3-A, let us consider that we are counting the number of cycles, in the wave packet starting at time t

3.2 |

3.3 |

3.4 |

DnDt
= 1 cycle |
3.5 |

DEDt
= h cycles·h |
3.6 |

Eq. 3.6
is
a form of uncertainty when cycles are counted. Before concluding, let
us
examine the case of radians.

In the
experiment
described, instead of counting cycles, the detector could monitor the
phase
of the incoming wave, counting an integer number of radians. The
radians
are recorded and counted until the time Dt
has
elapsed. It is found that y radians are counted at the end of the time
interval Dt. Again, after the last radian
has
been counted, an unknown fraction of a radian has started before the
end
of Dt. Therefore, the number of radians in
the
wave packet is between number y and y+1.

Since one
cycle equals 2p radians, the maximum
possible
frequency is:

3.7 |

3.8 |

3.9 |

3.10 |

3.11 |

3.12 |

3.13 |

3.14 |

3.15 |

3.16 |

With respect to the resolution, the method of counting an integer number of radians is equivalent to the Fourier transform. The Fourier transform however gives supplementary information because one can get the shape of the distribution while the method of counting the integer number of radians (or any determined angle) gives an average value of DE. Other instruments such as the lock-in amplifier can show the frequency distribution experimentally. Of course, the mathematical distribution given by the Fourier transform has an absolute shape (Fig. 3-B). It is the absolute shape of the resolution given by a spectroscope.

Since Heisenberg's resolution is mathematically identical to the result obtained when we are counting an integer number of radians (Eq. 3.11), we must conclude that the resolution of one radian corresponds to Heisenberg's criteria of resolution. This resolution of one radian is exactly equal to the loss of information that is intrinsic to the use the Fourier transform when the phase is neglected. In other words, Heisenberg's criteria of resolution has been set to be mathematically equal to the loss of information when we use a spectroscope. It is certainly

The angle of 1 radian, 1/2p cycle or 57.3 degrees, used by Heisenberg is certainly not an absolute or physical criterion. He could have decided to count cycles (or any other unit). In that case, the so-called fundamental uncertainty relationship would be different as shown in equations 3.6 and 3.16 In fact, it was neither rational nor coherent for Heisenberg to use the unit

Whatever the interpretation is, the relation should be written:

DEDt
= kh |
3.17 |

The uncertainty relationship claimed is

3.18 |

3.19 |

**3-8 Irrelevance of the
Planck
Constant h to Photon Energy.**

It is believed that following Eq. 3.1, the Planck constant determines the amount of uncertainty of energy of the photon. On the contrary, one can show that the uncertainty is not a function of the Planck constant. Let us divide both sides of Eq. 3.1 by h and let us substitute E by its value hn. Of course, such a division on both sides of the equation does not change the nature of the equation. This gives:

3.20 |

3.21 |

**3-9 Hidden Units.**

Another observation is related to hidden units. In practice, the unit

Finally, from Eq. 3.11, we can see now that the units of DEDt = h/2p are: "radian·Joule·s." and not simply "Joule·s." To avoid confusion, it is essential to indicate clearly that the units are "radian·Joule·s".

So, there is a hidden and arbitrary criterion of resolution in Heisenberg's uncertainty relationship which means that there is no fundamental physical principle involved. Heisenberg's relationship (Eq. 3.11) which corresponds to a criterion of one radian, is just as arbitrary as Eq. 3.16, which corresponds to a criterion of one degree (or 1/57.3 radian), as Eq. 3.6, which corresponds to a criterion of one cycle. Since the Heisenberg uncertainty relationship is an arbitrary criterion that is one of the most important aspects of the Copenhagen Interpretation [3.1]we must realize that it is necessary to reevaluate the significance of Quantum Mechanics [3.1].

*3-10 The Relationship
E
= hn*

In physics,
we have the relationship:

E = hn |
3.22 |

3.23 |

3.24 |

**References**

3.1 Cramer, John G., "The Transactional Interpretation of Quantum Mechanics", in

3.2 Heisenberg, Werner,

3.3 Lew, H., Marmet, N., Marshall, M. D., McKellar, A. R. W., Nichols, G. W., "A Compact Automatic Wavemeter for Use with Tunable Infrared Diode Lasers", in

3.4 Hall, J. L., Lee, S. A., "Interferometric Real-time Display of CW Dye Laser Wavelength with Sub-Doppler Accuracy", in

3.5 Schiff, Leonard I.,

3.6 Van Name, F. W. Jr.,

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