Observer’s Mathematics Approachto Quantum Mechanics
Dmitriy Khots, Ph.D. Boris Khots, [email protected] [email protected]
Observer’s Mathematics Approach to Quantum Mechanics – p. 1/45
Background• W − set of all real numbers.• Wn − set of all finite decimal fractions of length2n.
• Wn = {⋆ · · · ⋆︸ ︷︷ ︸n
. ⋆ · · · ⋆︸ ︷︷ ︸n
}.
• Concept ofobservers.
Observer’s Mathematics Approach to Quantum Mechanics – p. 2/45
Observers• All observers are naive.• Eachthinks that he lives inW , but• Eachdeals with Wn, so calledWn-observer.• Each sees more naive observers, i.e.,• Wn1
-observer can identify thatWn2-observer is
naive ifn1 > n2.
Observer’s Mathematics Approach to Quantum Mechanics – p. 3/45
Observers - More Specifically• Assumen1 > n2, then• ⋆→ ∞ for Wn2
-observer means⋆→ 10n2 forWn1
-observer.• ⋆→ 0 for Wn2
-observer means⋆→ 10−n2 forWn1
-observer.• Forn1 > n2 > · · · > nk, visual example:
Observer’s Mathematics Approach to Quantum Mechanics – p. 4/45
Arithmetic - Addition & Sub-traction
• For c = c0.c1...cn, d = d0.d1...dn ∈Wn
c±n d =
{c± d, if c± d ∈Wn
not defined, ifc± d /∈Wn
write ((... (c1 +n c2) ...) +n cN ) =N∑
i=1
nci for
c1, ..., cN iff the contents of any parenthesis are inWn.
Observer’s Mathematics Approach to Quantum Mechanics – p. 5/45
Arithmetic - Multiplication• For c = c0.c1...cn, d = d0.d1...dn ∈Wn
c×n d =n∑
k=0
nn−k∑
m=0
n0. 0...0︸︷︷︸
k−1
ck · 0. 0...0︸︷︷︸m−1
dm
wherec, d ≥ 0, c0 · d0 ∈Wn, 0. 0...0︸︷︷︸
k−1
ck · 0. 0...0︸︷︷︸m−1
dm
is the standard product, andk = m = 0 means that0. 0...0︸︷︷︸
k−1
ck = c0 and0. 0...0︸︷︷︸m−1
dm = d0. If eitherc < 0
or d < 0, then we compute|c| ×n |d| and definec×n d = ± |c| ×n |d|, where the sign± is definedas usual. Note, if the content of at least oneparentheses (in previous formula) is not inWn,thenc×n d is not defined.
Observer’s Mathematics Approach to Quantum Mechanics – p. 6/45
Arithmetic - Division• Division is defined to be
c÷nd =
{r, if ∃!r ∈ Wn, r ×n d = c
not defined, if no suchr exists or not !
Observer’s Mathematics Approach to Quantum Mechanics – p. 7/45
Arithmetic - General• The arithmetic coincides with standard if the
numbers are away fromWn borders.• If the borders aretouched, then other properties
arise.• Mathematics based on idea of observers, given
these arithmetic rules:• Observer’s Mathematics− Mathematics of
Relativity.• For more info, visitwww.mathrelativity.com.
Observer’s Mathematics Approach to Quantum Mechanics – p. 8/45
Philosophical Aspects• Two great Russian Geometers Rashevsky (MSU) and Norden
(KSU) discussed infinite-dimensional Lie Groups.
• One of the authors was present and heard Norden’s remark: "Yes,
but infinity does not exist".
• Possible misuse of ordinary Differential Geometry concepts such
as the limit, derivative, and integral.
• These instruments provide an advanced mathematical apparatus,
with possibly faulty assumptions:
• Space continuity
• Functions being continuous and differentiable
• These methods and calculations may be erroneous since arbitrarily
small or arbitrarily large numbers may not exist.Observer’s Mathematics Approach to Quantum Mechanics – p. 9/45
Geometrical Aspects• Lines, planes, or geometrical bodies, etc exist only
in our imagination.• These shapes cannot be approached with an
arbitrary accuracy due to instrument inaccuracy.• Avoiding infinity, Hilbert had created Geometrical
bases practically without the use of continuityaxioms: Archimedes and completeness.
• We find similar problems occurring in Arithmetic,and in entire Mathematics, since it is "arithmetical"in nature.
Observer’s Mathematics Approach to Quantum Mechanics – p. 10/45
Euclidean and Lobachevsky Ge-ometriesTheorem 1. Given a point (0, b) and a line l0, there isa line y = k ×n x+n b which is parallel to l0 inLobachevsky sense iff |b| ≥ 1, and in case |b| < 1, wewould only have parallel lines in Euclidean sense.
b k Notes
0, 0.01,. . . , 0.99 0 ∃ unique line through(0, b) parallel tol0
in Euclidean sense.
1, 1.01, . . . , 1.98 0.01 ∃ lines through(0, b) parallel tol0 in
Lobachevsky sense; given two values ofb
the lines are Euclidean parallel
1.99, 1.00, . . . , 2.97 0.02 ...
2.98, 2.99,. . . , 3.96 0.03 ...
. . . . . . ...
Observer’s Mathematics Approach to Quantum Mechanics – p. 11/45
Euclidean and Lobachevsky Ge-ometries
(0,b)
(-99.99,0.01)
(-99.99,-0.01) (99.99,-0.01)
(99.99,0.01)
0 l
(0,-b)
Observer’s Mathematics Approach to Quantum Mechanics – p. 12/45
Number Theory and Observer’sMathematics
• Fermat’s Problem Analogy.• TheoremFor anyn,Wn, n ≥ 2 and for anym ∈ Wn ∩ Z with m > 2, there exists positivea, b, c ∈Wn, such thatam +n b
m = cm.• Wherexm = (. . . (x×n x) × x . . .)×n) for anyx ∈Wn.
Observer’s Mathematics Approach to Quantum Mechanics – p. 13/45
Number Theory and Observer’sMathematics
• Mersenne’s Problem.• Mersenne’s numbers are defined asMk = 2k − 1,
with k = 1, 2, . . ..• The following question is still open: is every
Mersenne’s number square-free?• Theorem (Analogy of Mersenne’s numbers
problem). There exist integersn, k ≥ 2,Mersenne’s numbersMk, with {k,Mk} ∈Wn, andpositivea ∈Wn, such thatMk = a2.
Observer’s Mathematics Approach to Quantum Mechanics – p. 14/45
Number Theory and Observer’sMathematics
• Fermat’s Numbers Problem.• Fermat’s numbers are defined asFk = 22k + 1,k = 0, 1, 2, . . .. T
• The following question is still open: is everyFermat’s number square-free?
• Theorem (Analogy of Fermat’s numbersproblem). There exist integersn, k ≥ 2, Fermat’snumbersFk, {k, Fk} ∈Wn, and positivea ∈Wn,such thatFk = a2.
Observer’s Mathematics Approach to Quantum Mechanics – p. 15/45
Number Theory and Observer’sMathematics
• Tenth Hilbert Problem.• TheoremFor any positive integersm,n, k ∈Wn,n ∈Wm,m > log10(1 + (2 · 102n − 1)k), from thepoint of view of theWm−observer, there is analgorithm that takes as input a multivariablepolynomialf(x1, . . . , xk) of degreeq in Wn andoutputs YES or NO according to whether thereexista1, . . . , ak ∈Wn such thatf(a1, . . . , ak) = 0.
Observer’s Mathematics Approach to Quantum Mechanics – p. 16/45
Observer’s Math Meets Art• Consider all segments from the origin to any point
inside2 × 2 square centered at origin.• "Nadezhda" Effect: some segments do not exist,
due to nonexistence of their length, given by√
x2 +n y2 =√
(x×n x) +n (y ×n y)
Observer’s Mathematics Approach to Quantum Mechanics – p. 17/45
Art in W3
Observer’s Mathematics Approach to Quantum Mechanics – p. 18/45
Nadezhda Effect Theorem• Nadezhda Effect Theorem.• TheoremFor any positive integern andWn,
consider the planeWn×Wn = {(x, y)}, x, y ∈Wn
with standard Euclidean metricd2 ((x1, y1), (x2, y2)) = (x1 − x2)
2 + (y1 − y2)2.
Next, consider any liney = k ×n x, withy, k, x ∈ Wn. Then there is some point(x0, y0) = (x0, k ×n x0) ∈Wn ×Wn such thatd((x0, y0), (0, 0)) does not exist.
Observer’s Mathematics Approach to Quantum Mechanics – p. 19/45
Physical Aspects• Dynamics of a system change when the scale is
changed at which the system is probed.• For fluids, entire differenttheories are needed to
describe behavior.• At ∼ 1 cm - classical continuum mechanics (Navier-Stokes equations).
• At ∼ 10−5 cm - theory of granular structures.
• At ∼ 10−8 cm - theory of atom (nucleus + electronic cloud).
• At ∼ 10−13 cm - nuclear physics (nucleons).
• At ∼ 10−13 − 10−18 cm - quantum chromodynamics (quarks).
• At ∼ 10−33 cm - string theory.
• Mathematical apparatus applied to math models ofphysical processes can operate with any numbers,which creates room for error.
Observer’s Mathematics Approach to Quantum Mechanics – p. 20/45
Derivatives• In Wn-observer,y = y(x) is called differentiable atx = x0 if there existsy′(x0) = lim
x→x0,x6=x0
y(x)−y(x0)x−x0
• What does the above statement mean from point ofview ofWm-observer withm > n?
• |(y(x) −n y(x0)) −n (y′(x0) ×n (x−n x0))| ≤0. 0 . . . 01︸ ︷︷ ︸
n
whenever|y(x) −n y(x0)| = 0. 0 . . . 0yl︸ ︷︷ ︸
l
yl+1 . . . yn
and|(x−n x0)| = 0. 0 . . . 0xk︸ ︷︷ ︸
k
xk+1 . . . xn for
1 ≤ k, l ≤ n, andxk - non-zero digit.
Observer’s Mathematics Approach to Quantum Mechanics – p. 21/45
Derivatives• Theorem 1From the point of view of aWm-observer a derivative calculated by a
Wn-observer (m > n) is not defined uniquely.
• Theorem 2From the point of view of aWm-observer withm > n, |y′(x0)| ≤ Cl,kn ,
whereCl,kn ∈ Wn is a constant defined only byn, l, k and not dependent ony(x).
• Theorem 3From the point of view of aWm-observer, when aWn-observer (with
m > n ≥ 3) calculates the second derivative:
y′′(x0) = limx1→x0,x1 6=x0,x2→x0,x2 6=x0,x3→x1,x3 6=x1
y(x3)−y(x1)(x3−x1)
−y(x2)−y(x0)
x2−x0
x1 − x0
we get the following unequality:
(|x2 −n x0| ×n |x3 −n x1|) ×n |x1 −n x0| ≥ 0. 0 . . . 01︸ ︷︷ ︸
n
provided thaty′′(x0) 6= 0.
Observer’s Mathematics Approach to Quantum Mechanics – p. 22/45
Physical Interpretation• Hypothesis 1Theorem 1 could offer an explanation of why physical speed (or acceleration)
is not uniquely defined and, from the point of view of a measurement system (observer), it is
possible to consider speed (or acceleration) as a random variable with distribution
dependend on the measurement system. Letv be the speed with
v = v0.v1 . . . vn−k + ξn,km whereξn,k
m ∈ {0. 0 . . . 0︸ ︷︷ ︸
n−k
vn−k+1 . . . vn} - random variable,
m > n, and the distribution function isF n,km (x) = P (ξn,k
m < x).
• Hypothesis 2Theorem 1 could offer an explanation of why the speed of any physical body
cannot exceed some constant, (the speed of light, for example). Independence of this
constant on explicit expression of space-time function could offer an explanation of why the
speed of light does not depend on an inertial coordinate system.
• Hypothesis 3Theorem 2 could offer an explanation of the various uncertainty principles,
when a product of a finite number of physical variables has to be not less than a certain
constant. This can be seen not just from consideration of second derivatives, but of any
derivative.
• Hypothesis 4Theorems 1, 2, and 3 combined may provide an insight into the connection
between classical and quantum mechanics.
Observer’s Mathematics Approach to Quantum Mechanics – p. 23/45
Cauchy - Kowalevski TheoremThe Cauchy-Kowalevski theorem is the main local existence and uniqueness theorem for analytic
partial differential equations associated with Cauchy initial value problems. A special case was
proved by Augustin Cauchy and the full result by Sophie Kowalevski. The first order
Cauchy-Kowalevski theorem is about the existence of solutions to a system ofm differential
equations inn dimensions when the coefficients are analytic functions. The theorem and its proof
are valid for analytic functions of either real or complex variables.
Let K denote either the fields of real or complex numbers and letV = Km andW = Kn. Let
A1, . . . , An−1 be analytic functions defined on some neighborhood of(0, 0) in V × W and
taking values in them × m matrices, and letb be an analytic function with values inV on the
same neighborhood. Then there is a neighborhood of0 in W on which the quasilinear Cauchy
problem
∂xnf = A1(x, f)∂x1
f + . . . + An−1(x, f)∂xn−1
f + b(x, f)
with initial conditionf(x) = 0 on the hypersurfacexn = 0 has a unique analytic solution
f : V → W near0.
Observer’s Mathematics Approach to Quantum Mechanics – p. 24/45
Cauchy - Kowalevski TheoremLewy’s example shows that the theorem is not valid for all smooth functions. The theorem can
also be stated in abstract (real or complex) vector spaces. LetV andW be finite-dimensional real
or complex vector spaces, withn = dimW . Let A1, . . . , An−1 be analytic functions with
values inEnd(V ) andb an analytic function with values inV , defined on some neighborhood of
(0, 0) in V × W . In this case, the same result holds.
The higher-order Cauchy-Kowalevski theorem can be stated as follows. If F andfj are analytic
functions near0, then the non-linear Cauchy problem∂kt h = F (x, t, ∂j
t , ∂αx h), wherej < k
and|α| + j ≤ k, with initial conditions∂jt h(x, 0) = fj(x), with 0 ≤ j < k, has a unique
analytic solution near0. This follows from the first order problem by considering thederivatives
of h appearing on the right hand side as components of a vector-valued function.
Analysis of concepts such as Free Wave equation, Schrodinger equation, two-slit interference,
wave-particle duality for single photons, uncertainty principle, Airy and Korteweg-de Vries
equations, and Schwarzian derivative shows that in Observers’s Mathematics Cauchy-Kowalevski
theorems become invalid. Instead, we have stochastic properties of partial (and ordinary)
differential equations, both linear and non-linear.
Observer’s Mathematics Approach to Quantum Mechanics – p. 25/45
Newton Equation• Let F (x, t) = m ×n x. Then we have the following
• Theorem If the body with massm = m0.m1 . . . mkmk+1 . . . mn, with m ∈ Wn,
moves with accelerationx, |x| = x0.x1 . . . xlxl+1 . . . xn, with x ∈ Wn, and
m0 = m1 = . . . = mk = 0, mk+1 6= 0, k < n, x0 = x1 = . . . = xl = 0, l < n,
k + l + 2 ∈ Wn, n < k + l + 2 ≤ q, thenF (x, t) = 0.
• Corollary If l = n − 1 andk = 0, i.e.,m < 1, thenF (x, t) = 0.
• Theorem If l = n − 1 andxn 6= 0 then|F (x, t)| < 9.
• Theorem If m0 ≥ 9 . . . 9︸ ︷︷ ︸
p
, 0 < p ≤ n, x0 ≥ 9 . . . 9︸ ︷︷ ︸
r
, 0 < r ≤ n, n < p + r ≤ q, then
there is no forceF (x, t), such thatF (x, t) = m ×n x.
Observer’s Mathematics Approach to Quantum Mechanics – p. 26/45
ComplexifiedWn
• Complexification ofWn is defined byCWn = {x+ iy}, x, y ∈Wn.
• CWn has standardWn addition and multiplication .• (x1 + iy1)+n (x2 + iy2) = (x1 +n x2)+ i(y1 +n y2).• (x1 + iy1) ×n (x2 + iy2) =(x1 ×n x2 −n y1 ×n y2) + i(x1 ×n y2 +n y1 ×n x2)
Observer’s Mathematics Approach to Quantum Mechanics – p. 27/45
Schrodinger Equation• Consider the following:−(~×n ~) ×n Ψxx +n ((2 ×n m) ×n V ) ×n Ψ =
= i((2 ×n m) ×n ~)Ψt , whereΨ = Ψ(x, t), ~ is the Planck’s Constant,
~ = 1.054571628(53) × 10−34 m2kg/s. Let Ψ = Ψa + iΨb.
• TheoremLet 36 < n < 68, m = m0.m1 . . . mkmk+1 . . . mn, with m ∈ Wn,
m0 = m1 = . . . = mk = 0, mk+1 6= 0, k + 35 < n, V = 0, then
Ψt = Ψ0t .Ψ1
t . . . ΨltΨ
l+1t . . . Ψn
t andΨ0t = . . . Ψl
t = 0, Ψl+1t , . . . , Ψn
t are free and in
{0, 1, . . . , 9}, wherel = n − k − 36, i.e.,Ψt is a random variable, with
Ψt ∈ {(0.
n︷ ︸︸ ︷
0 . . . 0︸ ︷︷ ︸
l
∗ . . . ∗)}, where∗ ∈ {0, 1, . . . , 9}. Ψt = Ψat + iΨb
t
• Corollary Let 36 < n < 68, m = m0.m1 . . . mkmk+1 . . . mn, with m ∈ Wn,
m0 = m1 = . . . = mk = 0, mk+1 6= 0. Also, letV = υ0.υ1 . . . υsυs+1 . . . υn, with
V ∈ Wn, υ0 = υ1 = . . . = υs = 0, υs+1 6= 0, with k + 35 < n andk + s + 2 > n,
thenΨt = Ψ0t .Ψ1
t . . . ΨltΨ
l+1t . . . Ψn
t andΨ0t = . . . Ψl
t = 0, Ψl+1t , . . . , Ψn
t are free and
in {0, 1, . . . , 9}, wherel = n − k − 36, i.e.,Ψt is a random variable, with
Ψt ∈ {(0.
n︷ ︸︸ ︷
0 . . . 0︸ ︷︷ ︸
l
∗ . . . ∗)}, where∗ ∈ {0, 1, . . . , 9}. Note, in these theorems,Ψt means
bothΨat andΨb
t ; Ψxx means bothΨaxx andΨb
xx.
Observer’s Mathematics Approach to Quantum Mechanics – p. 28/45
Dirac Equations for Free Elec-tron
• −m0cψ2 = ~
(∂ψ
1
∂x3 +∂ψ
1
∂x0 +∂ψ
2
∂x1 + i∂ψ
2
∂x2
)
• m0cψ1 = ~
(∂ψ
1
∂x1 − i∂ψ
1
∂x2 −∂ψ
2
∂x3 +∂ψ
2
∂x0
)
• −m0cψ2 = ~
(∂ψ1
∂x3 + ∂ψ1
∂x0 + ∂ψ2
∂x1 − i∂ψ2
∂x2
)
• m0cψ1 = ~
(∂ψ1
∂x1 + i∂ψ1
∂x2 −∂ψ2
∂x3 + ∂ψ2
∂x0
)
• Wherex0 = Ct, x1 = x, x2 = y, x3 = z, ~ = h2π , h
is the Planck Constant, and~ = 1.054 . . . × 10−34
m2kg/s, C is the speed of light, andψ1, ψ2, ψ1, ψ2are the spinors.
Observer’s Mathematics Approach to Quantum Mechanics – p. 29/45
Dirac Equations In Observer’sMath
• ψ1 = ψa1 + iψb1• ψ2 = ψa2 + iψb2• ψ1 = ψa
1+ iψb
1
• ψ2 = ψa2
+ iψb2
• −(m0 ×n c) ×n ψa2 =
~ ×n
((((∂ψa
1
∂x3 +n∂ψa
1
∂x0
)
+n∂ψa
2
∂x1
)
−n∂ψb
2
∂x2
))
• −(m0 ×n c) ×n ψb2 =
~ ×n
((((∂ψb
1
∂x3 +n∂ψb
1
∂x0
)
+n∂ψb
2
∂x1
)
+n∂ψb
2
∂x3
))
• The other three equations are done similarly.
Observer’s Mathematics Approach to Quantum Mechanics – p. 30/45
Dirac Equations in Observer’sMath - Theorem
• Theorem If m0 is small enough such thatm0 ×n c = 0 then
•
((∂ψa
1
∂x3 +n∂ψa
1
∂x0
)
+n∂ψa
2
∂x1
)
−n∂ψb
2
∂x2 = 0.
n︷ ︸︸ ︷
0 . . . 0︸ ︷︷ ︸n−35
∗ . . . ∗
and
•
((∂ψb
1
∂x3 +n∂ψb
1
∂x0
)
+n∂ψb
2
∂x1
)
+n∂ψb
2
∂x3 = 0.
n︷ ︸︸ ︷
0 . . . 0︸ ︷︷ ︸n−35
∗ . . . ∗
• where any∗ ∈ {0, 1, . . . , 9} and is random.• Same statement is correct for the other three
equations.
Observer’s Mathematics Approach to Quantum Mechanics – p. 31/45
Two-Slit Interference• Let Ψ1 wave from slit 1.• Let Ψ2 wave from slit 2.• Ψ = Ψ1 + Ψ2 (with V = 0 in Schrodinger
equation).• TheoremThe probability ofΨ a wave is 0.45.
Observer’s Mathematics Approach to Quantum Mechanics – p. 32/45
Wave-Particle Duality for SinglePhotons
• λ×n (m×n v) = h whereh is the Planck constant.• Theorem If v is small enough, thenλ is a random
variable.
Observer’s Mathematics Approach to Quantum Mechanics – p. 33/45
Uncertainty Principle• ∆p×n ∆x = h
• Theorem• If ∆p is small enough, then∆x is a random
variable.• If ∆x is small enough, then∆p is a random
variable.
Observer’s Mathematics Approach to Quantum Mechanics – p. 34/45
Geodesic Equation• Consider the following:
xi +n
∑
jn∑
knΓi
jk×n (xj ×n xk) = 0
with j, k ∈ G.
• Theorem If xp = xp0 .xp
1 . . . xplxp
l+1 . . . xpn, with p ∈ G, xp
0 = xp1 = . . . = xp
l= 0,
0 ≤ l ≤ n, n < 2l ≤ q, then we havexi = 0, i.e., the geodesic curve is a line.
Observer’s Mathematics Approach to Quantum Mechanics – p. 35/45
Free Wave Equation• utt −n ((c×n c) ×n uxx) = 0
• Theorem 1If c anduxx are small enough, thenutt = 0 .
• Theorem 2If c anduxx are large enough, thenuttdoes not exist.
Observer’s Mathematics Approach to Quantum Mechanics – p. 36/45
Airy and Korteweg-de VriesEquations
• ut +n uxxx = 0
• (ut +n uxxx) +n (6 ×n (u×n ux)) = 0
• Theorem If u andux are small, then Airy equationand Korteweg-de Vries equations have the samesolutions.
Observer’s Mathematics Approach to Quantum Mechanics – p. 37/45
Probability in Quantum TheoryThe Hamiltonian is the Legendre transform of the Lagrangian:
H(qi, pi, t) =K∑
j=1
qjpj −L(qi, qi, t)
for i = 1, ..., K.
If the transformation equations defining the generalized coordinates are independent oft, and the
Lagrangian is a product of functions (in the generalized coordinates) which are homogeneous of
order 0, 1 or 2, then it can be shown thatH is equal to the total energyE = T + V .
Each side in the definition ofH produces differential:
dH =K∑
i=1
[∂H
∂qi
dqi +∂H
∂pi
dpi
]
+∂H
∂tdt =
=K∑
i=1
[
qidpi + pidqi −∂L
∂qi
dqi −∂L
∂qi
dqi
]
−∂L
∂tdt
Observer’s Mathematics Approach to Quantum Mechanics – p. 38/45
Probability in Quantum Theory• Theorem 1If p, q ∈ W2, from m−observer point of view withm > 8, then
P ((p +2 ∂p) ×2 (q +2 ∂q) −2 p ×2 q = p ×2 ∂q +2 q ×2 ∂p) = 0.8
where P is the probability.
• Theorem 2If p, q ∈ Wn, from m−observer point of view withm > 4n, then
P ((p +n ∂p) ×n (q +n ∂q) −n p ×n q = p ×n ∂q +n q ×n ∂p) = Pm,n < 1
wherePm,n is the probability dependent onm andn.
• Theorem 3If p, q ∈ Wn, from m−observer point of view withm > 4n, then
P (dH ≡ d(p ×n q −n L(q, q, t)) =
= q×n ∂p−n∂L
∂q(q, q+n ∂q, t+n ∂t)×n ∂q−n
∂L
∂t(q, q+n ∂q, t)×n ∂t) = Pm,n < 1
• Theorem 4If p, q ∈ Wn, from m−observer point of view withm > 4n, then
P (dH ≡ d(p ×n q −n L(q, q, t)) =
= q ×n ∂p −n∂L
∂q(q, q, t) ×n ∂q −n
∂L
∂t(q, q, t) ×n ∂t) = Pm,n,L < 1
wherePm,n,L is the probability dependent onm, n, andL.
Observer’s Mathematics Approach to Quantum Mechanics – p. 39/45
Probability in Quantum Theory• Theorem 5(K−bodies solution) Ifp, q ∈ Wn from m−observer point of view with
m ≥ log10
((2 × 102n − 1)2k + 1
)then
P
(
dH =K∑
i=1
(qi ×n ∂pi −n∂L
∂qi
(qi, qi, t) ×n ∂qi −n∂L
∂t(qi, qi, t) ×n ∂t
)
= Pm,n,L,K < 1
wherePm,n,L,K is the probability dependent onm, n, L, andK.
• Theorem 6Pn,m,K → 0 whenK → ∞, with m, n are the same fromm−Observer’s point
of view
• Theorem 7Let p ∈ [a1, b1], q ∈ [c1, d1] and[a1, b1] ×n [c1, d1] = Wn. With these
conditions, let probability that LHS = RHS bePn,m,K,[a1,b1],[c1,d1]. Also, letp ∈ [a2, b2],
q ∈ [c2, d2] and[a1, b1] ×n [c1, d1] = Wn. With these conditions, let probability that LHS
= RHS bePn,m,K,[a2,b2],[c2,d2]. ThenPn,m,K,[a1,b1],[c1,d1] = Pn,m,K,[a2,b2],[c2,d2]
• Theorem 8Pn1,m,K = Pn2,m,K with m, K are the same fromm−Observer’s point of
view andm > 4Kn1 andm > 4Kn2
Observer’s Mathematics Approach to Quantum Mechanics – p. 40/45
Lagrange Function of Free Ma-terial PointConsider simplest example of free movement of a material point with respect
to an inertial coordinate system. The Lagrange function in this case depends
only on the square of the velocity vector. To determine this dependence, we
will use Galileo’s principle of relativity. If inertial coordinate systemI is
moving relative to inertial coordinate systemI ′ with an infinitely small
velocity ǫ, thenv′ = v + ǫ. Since movement equations must have the same
form in all coordinate systems, then under this transformation the Lagrange
functionL(υ2) becomesL′, but will only differ fromL(υ2) by the full
derivative of coordinate and time function.
We then have the following:
L′ = L(υ′2) = L(υ2 + 2vǫ+ ǫ2)
Decomposing the above expression into a series of powers ofǫ and ignoring
infinitesimals of higher order, we get the following:
L(υ′2) = L(υ2) +∂L
∂υ22vǫ
Observer’s Mathematics Approach to Quantum Mechanics – p. 41/45
Lagrange Function of Free Ma-terial PointThe second part of the right hand side of the above equation will be the full
derivative with respect to time only in cases when it linearly depends on
velocityv. Thus, ∂L
∂υ2 2vǫ is independent of velocity, i.e. the Lagrange
function is directly proportional to square of velocityL = m
2υ2, wherem is a
constant. From the fact that Lagrange function of this type satisfies Galileo’s
principle of relativity in case of infinitely small velocitytransformation, it
follows that it also satisfies the principle in case of finite velocityV of
coordinate systemK with respect toK′. In fact, we have the following:
L′ =m
2υ2 =
m
2(v + V )2 =
m
2υ2 + 2
m
2vV +
m
2V 2
Observer’s Mathematics Approach to Quantum Mechanics – p. 42/45
Lagrange Function of Free Ma-terial Point
• Theorem 9P ((a+n b) ×n (a+n b) = (a×n a+n 2 ×n (a×n b)) +n b×n b) < 1
• Theorem 10P (c×n (a+n b) = c×n a+n c×n b) < 1
• Theorem 11In classical mechanics,P (L = m
2v2) < 1
Observer’s Mathematics Approach to Quantum Mechanics – p. 43/45
On The Way to Feynman Inte-gration
• Leibniz Theoremddx
∫
y
f(x, y)dy =∫
y
∂∂xf(x, y)dy.
• Observer’s Mathematics Theorem
P
(
ddx
∫
y
f(x, y)dy =∫
y
∂∂xf(x, y)dy
)
=
Pf,x,y,n,m < 1
• Wherex ∈ X andy ∈ Y andPf,x,y,n,m is theprobability dependent onf, x, y, n,m.
Observer’s Mathematics Approach to Quantum Mechanics – p. 44/45
Bibliography• Seewww.mathrelativity.com, with over 25
papers published.
Observer’s Mathematics Approach to Quantum Mechanics – p. 45/45