## A Mathematical Model for Calculating the Mass of the Neutrino

#### I. Protons

1. Consider the proton as an elementary unit of quantity. The "value" of a proton then may be considered as unity, i.e., one.

2. Let us represent the proton's mass as a positive one (+1), and the proton's wavelength (it's Compton wavelength) as a negative one (-1).

3.a. from the 1986 CODATA Recommended Values of the Fundamental Physical Constants, we find the standard value of the proton's mass (mass_p) and the proton's wavelength (lambda_p) given as follows:

mass_p = 1.6726231 x 10^-24 gm
lambda_p = 1.32141 x 10^-13 cm

(taken).

3.b. Considering each value as unity, we may substitute the following:

mass_p = 1.6726231 x 10^-24 gm = +1
lambda_p = 1.32141 x 10^-13 cm = -1

4.a. With these premises, we may express units of mass (gm) and units of distance (cm) as follows:

gm = +1 / 1.6726231 x 10^-24 = 5.978633 x 10^23
cm = -1 / 1.32141 x 10^-13 = -0.7567673 x 10^13

4.b. Physically, we can think of these two equations as:

gm = 5.978633 x 10^23 (mass_p) = 5.978633 x 10^23 x (+1)
cm = 0.7567673 x 10^-13 (lambda_p) = 0.7567673 x 10^-13 x (-1)

5. In addition to mass and distance, let us now consider time as another fundamental unity. To do so, let us try to define time in terms of the same unit, the proton. Speed or momentum is distance divided by time, so if we can find a fundamental "speed" which we can equate with our fundamental distance (the Compton wavelength of the proton), then we will have all terms in our equation, and we will be able to develop a fundamental unit of time from these.

The "natural" unit of speed is the speed of light (c), a fundamental constant of nature. If distance/time = speed, then a fundamental unit of distance divided by a fundamental unit of time may equal the speed of light. If the premises developed above have any significance, we should be able to divide the wavelength of the proton (lambda_p) by a fundamental unit of time (t) to derive the speed of light (c). In symbolic terms,

lambda_p / t = c

or

t = lambda_p / c

Again using the CODATA values, we have

t = lambda_p / c = 1.32141 x 10^-13 cm / 2.9979246 x 10^10 cm/sec

and, therefore,

t = 4.407749 x 10^-24 sec

We will call this unit of time the "time of a proton," or time_p, or simply t_p. This result (t_p = 4.407749 x 10^-24 sec) approximates the experimentally determined time scale for nuclear interactions -- often stated roughly as 10^-24 sec.

6. Using this unit of time (t_p), we can state that one second equals a certain number of t's, i.e.,

sec = t_p / 4.407749 x 10^-24 = 2.2687317 x 10^23 t_p

7. Using these units of mass, distance and time, let us imagine a number line where negative numbers represent all distances and positive numbers represent all masses, each expressed in units of time. To reiterate,

gm = 5.978633 x 10^23 mass_p
cm = -0.7567673 x 10^13 lambda_p
sec = 2.2687317 x 10^23 t_p

8. In his book, A Brief History of Time, Stephen Hawking invokes the concept of "imaginary time" for the purposes of certain quantum mechanical calculations (at p. 134), and relativistic effects (p. 139). One knot which this concept cuts through is the idea of time running forward or backward (p. 143), which makes "imaginary time" particularly useful in physics calculations which themselves make no distinction between forward and backward time, despite our experience of the "arrow of time" which travels exclusively forward.

"Imaginary time," in this sense, is not "fanciful time," but merely time calculations which make use of "imaginary numbers" which incorporate the square root of minus one (sqrt(-1)) -- a useful mathematical device with an ancient pedigree.

Let us suppose that we may represent time itself as a "negative" quantity, related to the square root of minus one. In order to relate this quantity to our experience, we may deal with the absolute value of this quantity, so that "time" is the absolute value of the square root of minus one, or |sqrt(-1)|.

t = 4.407749 x 10^-24 sec = |sqrt(-1)|.

Rather than the absolute value, we also may set "t" as a positive number by representing it as follows:

t = -(sqrt(-1))

Since -sqrt(-1) = 1 / +(sqrt(-1), we may further state that

t = 1 / -t
t = -1 / t

and-

t = 1 / t

Finally, we see that this formula further yields the result

t^2 = -1

II. Neutrinos

9. Neutrinos, as we understand them, are integral to the weak force or interaction. Such interactions typically are known to take approximately 10^-10 seconds. Using just the units of mass, distance and time derived above, we may further derive a "time of the neutrino" (tn), as we did the "time of the proton" (t_p).

gm.cm. / sec = (5.978633 x 10^23) (-0.7567673 x 10^13) / (2.2687317 x 10^23)t

= -4.5244339 x 10^36 / 2.2687317 x 10^23 t

= 1.9942569 x 10^13 x (-1 / t)

= 1.9942569 x 10^13 (t)
(from paragraph 8, above)

= (1.9942569 x 10^13) (4.407749 x 10^-24 sec)
(from paragraph 5, above)

= 8.7901838 x 10^-11 sec

= 0.8790184 x 10^-10 sec

which is close to the "roughly 10^-10 sec" mentioned above for nuclear interactions (paragraph 5).

10. Because the neutrino is closely associated with these weak nuclear interactions, we will infer that this time interval is associated with the neutrino itself. Therefore, we will denominate this time interval as the "neutrino time," or t_n.

11. We saw a relationship between time and the mass of the proton, in that the "proton time" divided by itself equaled the "proton mass." (Mentioned above.) Thus, we may state

delta-t_p / delta-t_p = +1 = mass_p

Let us hypothesize that any time interval, divided by any other time interval, will similarly represent some mass, as a multiple or fraction of the mass of a single proton. (Here, we exclude "negative time" speculations.)

delta-t_x / delta-t_y = k(+1) = a mass

where k is a positive number, whole or fractional.

Thus, time divided by time equals a mass -- a mass multiple -- a multiple of the proton's mass, which is one.

12. Therefore, if the neutrino's time (t_n) is as set forth in paragraphs 9 and 10, above, we should be able to divide the neutrino time by itself to get a mass associated with the neutrino (mass_n). This would work as follows:

mass_n = (gm.cm. / sec) / sec

= gm.cm. / sec^2

which may also be expressed

(cm / sec^2) gm

13. With the values derived above for gm, cm, and sec, we can derive a mass associated with the neutrino as follows:

mass_n = gm.cm. / sec^2 = (5.978633 x 10^23)(-0.7565673 x 10^13) / (2.2687317 x 10^23 t)^2

= -4.5244339 x 10^36 / (5.1471435 x 10^46) (-1)
(from paragraph 8, where t^2 = -1)

= 0.8790184 x 10^-10

Expressed in grams, this yields

mass_n = n(gm) = 0.8790184 x 10^-10

and

n(5.978633 x 10^23) = 0.8790184 x 10^-10

and

n = 0.8790184 x 10^-10 / 5.978633 x 10^23 = 0.1470266 x 10^-33

or 1.470266 x 10^-34

Thus, this method for correlating time to mass yields a neutrino mass of 1.470266 x 10^-34 gm.

This is a very small value for the mass of a particle, being many orders of magnitude less than the mass of an electron, which is to be expected given the difficulty of determining whether the neutrino had any mass at all.

14. As reported in the New York Times (June 5, 1998), recent experiments at the Super-Kamiokande detector, while not yielding a value of any neutrino masses, do "suggest that the difference between the masses of muon neutrinos and other types of neutrinos [in the oscillations] is only about 0.07 electron volts (a measure of particle mass)."

Mass may be expressed in electron volts (eV) because mass and energy are equated through e = mc^2. According to the CODATA values (Table 4),

1 eV = 1.78266270 x 10^-36 kg, or 1.78266270 x 10^-33 gm.

Therefore, 0.07 eV = 1.24786389 x 10^-34 gm.

By equating a particle's mass with the time of its interactions, we come up with a number that is comparable to the experimentally determined oscillation mass of the neutrinos. Because of the uncertainty in the reported experiments, it appears that our calculated value is at least in the right ball park. As experiments are refined, I expect that the experimentally determined value will approach more closely the calculated value.