Dmytro Taranovsky

March 25, 2000

To facilitate making and
understandings of models, scientists have agreed on the definitions of many
physical quantities. Scientists know many physical quantities, which are
classified into basic and derived. Basic quantities are undefined (unknown), while
the derived quantities are defined in terms of the basic.^{[1]}
The basic quantities are length, time, mass, charge, temperature, and the
amount of substance. Although the basic quantities are undefined, the ratio of
basic quantities of the same type is defined.^{[1]}
For example, the
ratio of masses may be defined as the ratio of gravitational accelerations they
create at a fixed distance. A value of a basic quantity is written as a
product of the value of the quantity for a specified object (the value is
called unit) and a real number. Because the quantities are unknown, we can
apply dimensional analysis (essentially checking that the unknown part of a
physical quantity (unit) is in a proper form).

Since not defining quantities is
bad, as few quantities as possible should be left undefined. From all physical
quantities, the real number is the only truly basic and indefinable quantity.
^{[2]}
Other scalar physical quantities can and should be defined as real numbers
(since no good alternatives exist). Doing so requires a numerical
specification of the basic physical quantities. This is best achieved by
setting the fundamental physical values to 1. The values probably are **c**,
**G**, **ħ**, **e** (charge of proton), **k** (Boltzmann's
constant), and **N _{A}**.

When the definitions (and all formulas that follow from them) are not used, the dimensional analysis can be applied completely. Using any of the formulas, however, restricts the application of dimensional analysis.

Dmytro Taranovsky

March 25, 2000

Physical quantities are defined in each physical model. To make understanding of physical models simpler, physicists agreed on definition/perception of the physical quantities common to many models. Therefore, physicists define/introduce physical quantities and they should decide whether defining physical quantities as described in the New Perception of Physical Quantities is better than leaving the physical quantities undefined as currently is.

Defining physical quantities will simplify the calculations since instead of the fundamental (and widely used) physical constants you can use 1. Dimensional analysis can still be completely used in a particular problem unless 1 is used instead of 'the conventional value' of a fundamental physical constant. If 1 is used as value of some fundamental constants but not all, dimensional analysis will be restricted but valid. Therefore, if dimensional analysis is more important than using some of the fundamental physical constants as one, the dimensional analysis can be completely applied by not using 1 as a value of some fundamental physical constants. Thus, the New Perception of Physical Quantities is at least the same and sometimes better than the Classical Perception of Physical Quantities in this aspect.

Another aspect is how natural,
simple, and logical the perceptions are. The new perception is simpler than
the classical since c, G, ħ, e, k, and N_{A} do not complicate
it. The new perception is also more natural since it defines physical
quantities. Several other insights support this conclusion. For example,
energy and mass are identical from the physical point of view yet different in
the classical perception. Recall that Euclidean 4-dimensional space-time is a
set of all ordered quadruples of real numbers. It is natural to model our
universe by Euclidean space-time. The classical perception prevents it since
distance and time are not real numbers in it.

In chemistry, it is very convenient to say that the charge of electron is -1 so that chemists can easily pronounce and write the charges of ions. This is allowed in the New Perception of Physical Quantities but not in the classical.

The final aspect is the validity of physical models as mathematical theories. Since physical quantities are undefined in the classical perception, they cannot be used in any valid physical models (that is mathematical theories which correspond to some events in our world). This defeats the purpose of introducing physical quantities. In the New Perception of Physical Quantities, physical quantities can and should be used.

These arguments show that physicists should choose the New Perception of Physical Quantities instead of what they have now--the Classical Perception of Physical Quantities.

Dmytro Taranovsky

March 25, 2000

In the Classical Perception of
Physical Quantities, physical quantities are undefined simply because the
definitions do not exist. Some fallacies may, however, resemble definitions.
The first and perhaps the most important fallacy is "defining" a physical
quantity in terms of undefined quantities. If the "definition" is pursued by
trying to define the undefined terms, circular definitions will soon be
reached. For Example: "__M≡ar ^{2}/G, G≡6.67...*10^{-11}m^{3}/(kg*s^{2}),
kg≡ M for which ar^{2}
is 6.67m^{3}/s^{2}. M is mass; a is the gravitational
acceleration the body creates at distance r from the body.__" Another fallacy is
defining quantities other than the quantity whose definition has to be found.
Often, it is defining a quantity proportional to the given one such as the
quantity over its units. For example: "

**Exercise:**
Create the 3 fallacies for temperature. (If you do not know enough about
temperature, read several paragraphs before you do the exercise. In several
paragraphs, temperature will be defined.)

A stronger statement can be given in the Classical Perception of Physical Quantities: No physical quantities that have dimensions can be defined from a set of physical quantities not containing a component of the dimension. This is a direct implication of dimensional analysis--a quantity cannot be equal to a combination of quantities not containing a component (such as mass) of the quantity.

The New Perception of Physical
Quantities provides the framework for defining all physical quantities. Our
universe can be represented by a set of all ordered quadruples of real numbers,
(x, y, z, t). The New Perception of Physical Quantities provides the scaling
between this representation and our universe. Distance, D((x_{1},y_{1},z_{1},t_{1}),
(x_{2},y_{2},z_{2},t_{2})) ≡
((x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}).
Time, T((x_{1},y_{1},z_{1},t_{1}),(x_{2},y_{2},z_{2},t_{2}))
≡ |t_{2}-t_{1}|.
Proper distance is defined as sqrt(D^{2}-T^{2}) provided that
the D>T. Proper time is defined as sqrt(T^{2}-D^{2})
provided that the T>D. As in the Classical Perception of Physical
Quantities, r≡(x,y,z), v≡dr/dt,
a≡dv/dt.

According to the law of
universal gravitation, the product of *a* (curvature of space-time created
by the body) and D^{2} (D is distance to the body) for each body is
independent of location provided that D is much larger than the size of the
body and the body is not moving with a relativistic speed. Mass, M≡aD^{2} for a stationary object.
For a moving object, M≡γM_{o} where M_{o} (called
rest mass) is defined as mass of the object if it was stationary, and γ≡sqrt(1-v^{2}).
As in the Classical Perception of the Physical Quantities, p≡M*v,
F≡dp/dt, ρ≡dM/dV. E≡M, P≡dE/dt.

Temperature (T) of a monatomic ideal gas is defined to be 2/3*E, where E is the average translational kinetic energy of its molecules. Temperature of an object other than a monatomic ideal gas is defined to be the temperature that a monatomic ideal gas would possess if it would be in thermal equilibrium with the object.

Defining electric charge has some difficulties because of the incomplete information known. Perhaps, it should be defined separately in each model. The following is a reasonable definition. It is unclear, however, whether it is a definition or merely a specification that is valid only in some models. Let electric field (E) be defined as the negative of the force that acts on an electron at rest. Let charge, q≡ F/E. Anyway, the New Perception of Physical Quantities allows electric charge to be defined as an experimental quantity (that is determinable by experiment) in any appropriate model.

Some other quantities
can be defined as real numbers. The amount of substance is defined as the
number of molecules in the given sample. Thus, 1mole=6.0...*10^{23} and
N_{A}=6.0...*10^{23}/mol=1. Consequently, R=N_{A}*k=k=1.
Information that can be stored in a system that can have N independent states
is defined as log_{2}N.

Real numbers too need to
be defined. Several ways exist to do this. The following is a good way to
define real numbers: A real number (X) is defined to be an integer (m) or an
integer with infinitive sequence of 0 and 1 (f(n)) so that for at least one n,
f(n)=1, and there is no n such that f(k)=1 for every k>n. The restrictions
on f(n) are necessary to ensure that real numbers that are different by this
definition are unequal. X=m+Σ(2^{-n}f(n),
n,1, ∞). The equation is not
part of the definition of real numbers. It is simply used to connect the
definition to some operations that we know on real numbers.

**Solution to the exercise:** First, we will take
some steps in the right direction. Temperature of an object other than a
monatomic ideal gas is defined to be the temperature that a monatomic ideal gas
would possess if it would be in thermal equilibrium with the object. Consequently,
I will give fallacies for monatomic ideal gas. Let E be the average
translational kinetic energy of its molecules. Using the first fallacy, let T
be "defined' as 2/(3k)*E where k≡1.38...J/K
where K≡
1/273.16 of the temperature at which the triple point of water occurs. Using the second
fallacy, temperature is the number of Kelvin thermometers will measure.
(Although the fallacy is stated informally, it can be formalized to define
T/K.) Using the third fallacy, temperature is a measure of hotness or coldness
a body possesses.

Here I will explain how to convert a
physical quantity to the "dimensions" you wish it to have. To do so, you
multiply it by c^{l}G^{m}ħ^{n}e^{o}k^{p}N^{r}
(subscript 'A' is ommited from n) so that the dimensions are proper. Let the
original dimension be X and the needed one--Y. In effect, you need to solve
(Y/X)=c^{l}G^{m}ħ^{n}e^{o}k^{p}N^{r}.
The equation can be solved by separating the dimensions into components:
length, time, mass, electric charge, temperature, and the amount of substance.
Then you will get a separate equation for each component. Solving the system
of equations, you will obtain l, m, n, o, p, and r. In a few cases, you will
have to perform the procedure several times such as if the quantity is given as
a sum of 2 quantities with different dimensions.

**Exercise:**Convert the sentence
to the Classical Perception of Physical Quantities:

"Evidence shows that the most fundamental length, time, and mass are 1."

**Solution:**
length=1=c^{l}G^{m}ħ^{n}e^{o}k^{p}N^{r}.
Since no component of length, c, G, or ħ is electric charge, temperature
or amount of substance, 'o', 'p', and 'r' are 0. 1=c^{l}G^{m}ħ^{n}.
D=(D/T)^{l} (D^{3}/(M*T^{2})^{m} (M*D^{2}/T)^{n},
D^{1}T^{0}M^{0}=D^{l+3m+2n} T^{-l-2m-n}
M^{-m+n}, 1=l+3m+2n and 0=-l-2m-n and 0=-m+n, m=n and 1=l+5n and
0=-l-3n, 1=2n, n=1/2 and m=1/2 and 1=l+2.5, l=-3/2, length is c^{-3/2}G^{1/2}ħ^{1/2}.
The time =1=(distance 1)/c= c^{-5/2}G^{1/2}ħ^{1/2}.
The mass = 1=c^{l}G^{m}ħ^{n}e^{o}k^{p}N^{r}.
Since no component of mass, c, G, or h is electric charge, temperature, or the
amount of substance, o=0, k=0, and r=0. M=c^{l}G^{m}ħ^{n}.
M=(D/T)^{l} (D^{3}/(M*T^{2})^{m} (M*D^{2}/T)^{n},
D^{0}T^{0}M^{1}=D^{l+3m+2n} T^{-l-2m-n}
M^{-m+n}, 0=l+3m+2n and 0=-l-2m-n and 1=-m+n, 0=m+n and n=m+1, 2m+1=0,
m=-1/2, n=1/2, l=1/2, the mass is c^{1/2}G^{-1/2}ħ^{1/2}.
Thus, the sentence will be: "Evidence shows that the most important length is
c^{-3/2}G^{1/2}ħ^{1/2}, time -- c^{-5/2} G^{1/2}ħ^{1/2},
and mass -- c^{1/2}G^{-1/2}ħ^{1/2}."

**Exercise:**Find the formula for
proper length and proper time in the Classical Perception of Physical
Quantities.

**Answer:** Proper length is sqrt(D^{2}-c^{2}T^{2}).
Proper time is sqrt(D^{2}-c^{2}T^{2})/c.

**Exercise:** What is the value (using no units) of Coulomb's constant (8.988...m/F)?

**Answer:** 0.007297... .

Dimensional analysis forbids logs
from dimensional quantities. Otherwise, ln(2m^{3}) = ln(2*1m^{3})=ln
2+ln(1m^{3}). Thus ln(x) would have to be dimensionless. Then, x=e^{ln
x} would have to be dimensionless as well leading to contradiction if x
has dimensions. Thus, for example,

∫dV/V≠ ln V+C since ln is not
defined for dimensional quantities. This problem does not occur in the New
Perception of Physical Quantities, since all quantities are dimensionless.

Dimensions can be formally defined
as the sequence of exponents for each basic unit in a quantity. For example,
force has dimensions (1, 1, -2, 0, 0, 0) corresponding to mass^{1}*length^{1}*
time^{-2}*charge^{0}*temperature^{0}*(amount of
substance)^{0}.

The fundamental constants chosen to be one must have linearly independent dimensions. Otherwise, one constant could be expressed from the others causing 2 quantities with the same dimension but different values in SI units be equal--a contradiction. The number of constants must equal the number of dimensions so that all units could be expressed as a combination of these constants. The constants chosen above satisfy the properties.

Sometimes, something that looks like
a unit is really a part of the type of quantity. For example: "The distance
between the 2 genes on a chromosome is *a* morgans." "Morgans" specifies
distance leading to the quantity type "the probability of crossing over between
the 2 genes" and the value *a*. (Of course, units *are* parts of
physical quantities, 1m≠1.)

[1] Definitions are sometimes incomplete, may change with new insights, and may differ slightly depending on a scientist.

[2] Actually, real numbers can be defined in terms of integers. Integer is the basic quantity.