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Mathematical modelling frameworks for SMEP (state of the art) and GUT (proposed)
Dr. Klaus Braun May 29, 2017
Some challenges of Standard Model of Elementary Particles (SMEP)
Quantum gravity is a field of theoretical physics that seeks to describe the force of gravity according to the principles of quantum mechanics. Ground state (or vacuum) energy modeling is concerned about necessarily “existing” “empty space” energy in the absence of matter. Light consists of particles. The current of electrons increases with the increase of its frequency. In the same way it does for the current of light. But the current of light does not increase with the increase of the intensity (the "force" of the light). This phenomenon leads Einstein to the concept of photons with minimal
quantum energy. But photons have no mass; nevertheless it holds the Einstein equation 2cmE . In
addition the light is an electro-magnetic wave in the sense of Maxwell equations. The Higgs boson combines the existence of mass together with the action of the weak force. But why it provides especially to the quarks that much mass, is still a mystery. The state of the art assumption is, that quantum fluctuation was the first mover in the early state of the universe; the 4 Nature forces arise at a later state after inflation. Energy breaking & phase transition are the current concepts to explain mass generation out of purely massless “particles” (photons).
[BlD]. Preface: “There is a standard way to compute the field strength (or curvature) of a gauge potential (or connection). In case where )1(UG (or equivalent the group of rotations in the plane), the gauge
potential is essentially the 4-vector potential of electromagnetism and the field strength is the electromagnetic field. H. Weyl introduced the concept of gauge transformation and gauge invariance. Yang and Mills introduced gauges prescribing a point-dependent choice of isotopic spin axis. In this case the group is SU(2) (or equivalent, the unit of quaternions).The Yang-Mills model was a precursor to the apparently successful model of Weinberg/Salam for weak interactions. The mechanism of spontaneous symmetry breaking allows gauge fields to acquire mass (consider, e.g. the massive “intermediate vector bosons” in the Weinberg-Salam model). In spite of these refinements, the basic fact remains that the existence of gauge fields is a consequence of the existence of gauge-invariant action densities for particle fields. …. The action density is a measure of the superfluous manifestations of the fields involved. Nature obeys the principle of least action.”
[BlD] Bleecker D., Gauge Theory and Variational Principles, Dover Publications, Inc., Mineola, New York, 1981
Comparison table: current vs. proposed
Standard Model of Elementary
Particles (SMEP)
Grand Unified Theory (GUT)
Proposed mathematical model
Physical model interpretation
all “forces”, which interact between different particles, are due to the
interaction of related different quantum “types”, resp. fields;
Standard Model of Elementary
Particles by the Yang-Mills field
)2()3()1()2()3( xUSUxUxSUSU ,
no single force only “at” big bang and generation of 4 forces during inflation phase; 4 pseudo forces, only, which
are measurable actions as a consequence of vacuum state energy as defined per considered variational
PDE or PDO
Mathematical framework
additive Gauge fields concept per “force type” in combination with
variational principles
)1(UG : group of rotations in the
plane
)2(SUG : the unit of quaternions,
Grassman-valued fermion field
single vacuum energy concept for all “pseudo-force” types in combination
with variational principles per considered PDE/PDO
Hankel & Hilbert transforms
n/a IdHH 22
HH , self-adjoint, 2L isometric
Wave and wave package concept
“Dirac” function; its regularity depending from space
dimension
distributional Hilbert space element of
2/1H ,
the Hilbert space regularity is independent from the space
dimension and better than “Dirac function” regularity for any space
dimension
quantum state framework 0,2/01 NHHH 0,2/102/1 NHHH
0H polynomial system Hermite polynomials
n whereby
1 is the Gaussian function
Hilbert transformed Hermite
polynomials H
n :
0),( H
nn , 0)0(ˆ H
n , i.e. the
commutator 0, Px for any
convolution (singular integral) Pseudo-Differential Operator
002/1 HHH n/a
orthogonal projection operator
02/10 : HHP
2/12/10 ),(),( vuvuP 0Hv
whereby for 2/1 Hu
02/1 ),(),( vuvu 2/1 Hv ;
note: the least action principle is equivalent to operator norm
minimization problem
Hamiltonian operator(s)
sum of Hamiltonian operators per considered gauge field
combinations
EinsteinHiggsMillsYangDirac HHHHH
single Hamiltonian operator per considered variational PDE/PDO
Harmonic oscillator, Hamiltonian operator
2
2
2
xdx
dH
creation & annihilation operators with different
domains
H Hilbert transform operator
01: HHxdx
dA
,
01
*: HHx
dx
dA
0),( vuA
, 0
*),( vAu
0Hv
H self adjoint, eigenvalues real,
AA
*
11**
AAAAH
2/12/1:
HHxdx
da
,
2/12/1
*:
HHx
dx
da
2/1),(
vua , 2/1
*),(
vau
2/1 Hv
*,aa are self adjoint, because of
002/1 ),(),(),( vuHvuAvu
2/100 ),(),(),( vuvAuvHu
dependency of space dimension
yes no
space-time dimensions
10N 4-dimensional scalar field
41 mn
Poincare “conjecture” n/a applicable
Huygens’ principle; spherical waves for
temporal lines of space-time dimension
n/a Cauchy & radiation problem for wave
equation
particles vs. photons ratio 10106 particles 0H , photons
02/1 HH
Planck’s action quantum
smallest action quantum, which can be measured in the test space
0H
smallest action quantum, which can be measured in the test space
0H
“time”
eigen-time of a photon is zero, i.e. a photon dies not realize that times goes by; how a photon can act?
de Broglie time:
E
ht
R. Penrose, The Emperor’s New Mind:
“the time of our perception does not “really” flow in quite the linear forward-moving way that we perceive it to do.
The temporal ordering that we “appear” to perceive is, …, something that we impose upon our perceptions
in order to make sense of them in relation to the uniform forward time-progression of an external physical
reality”
Schrödinger momentum
operator
i
iP
:
22
2
1
2:
immH
0)( HPD : “eigen-functions” are
plane-waves ikxexf )( which are
not defined in 0H , i.e.
0H
2/1)( HPD : eigen-functions are
plane-waves ikxexf )( which are
defined in 0H , i.e.
0H
Harmonic quantum
oscillator (generation & annihilation operators)
)2
1(
2
1
2
1
2
222
aaqmm
pHosc
)2
1(
2
1 aaHosc
0),( oscH 0H
nn EnE ),2
1(:
)(2
_
2*
_aacH
2/1
_),( H
0H
“empty” vacuum
“a closed vacuum space cannot be diluted”
(causing inflation/mass generation challenges in the early state of the
universe)
no particles in a vacuum, “just”
02/1 HH photons/waves
Heisenberg uncertainty principle
per affected gauge field/Hamiltonian operator
valid in 2/1H
not valid in 0H
Schrödinger equation
classical mechanics relationship: continuity equation, Fourier waves
0
*
0
*2
0),(),(
2 gradgrad
imt
classical mechanics relationship
continuity equation, Calderón wavelets
2/1
*
2/1
*2
2/1),(),(
2
gradgrad
imt
0),(),(2
0
*
0
*2
2/1
imt
, 0H
Generation of mass of sub-atomic particles and
its mass value
energy breaking, phase transition; Lagrange (Klein-Gordon) density
and potential of the Higgs field , which is a kind of ether existing
throughout the universe
)())(( VDDLHiggs
22 )(:)( V ,
D covariante derivative
phase transition between 2/10 HH
and the closed sub-space 02/1 HH
Casimir, Lamb effect verification in test space 0H verification in test space
0H
Elementary particles and the options
0: HH
or
002/1: HHHH
Einstein-Podolsky-Rosen paradoxon
0HH ,
0H
gauge/Higgs/(hypothetical) graviton bosons with only symmetric Schrödinger wave function
solutions
0HH ,
0H
“matter” particles fermions with only anti-symmetric Schrödinger wave
function solutions
00 H
with 100
one therefore gets 1
000
2/1 HH
“wavelet” function/field space
0H
fermions (sub-) space = (matter) test space
002/100 HHH
with 1
00 ,
:2/1
0
one therefore gets
1
212
2/1i
ixe i
Single and double layer potential
Lebesgue integrals,
mass density
ds
ds
)(
and existing normal derivative with corresponding (mathematically
required) regularity assumptions
dn
dU
Stieltjes integrals,
Plemelj’s concepts of an mass element
sd
and related “current of a force” on the boundary through the boundary
element ds
0
)( dsdn
dUU
s
Quantum theory of radiation,
intensity = surface force density
ensityint
densityforcesurface __ densityenergy _
Pascalcm
Ncm
cm
N
32
regularity requirement/assumptions of an existing normal derivative with
exterior/interior domain w/o physical meaning
very poor regularity assumptions in the form
0
d
to ensure continuous single layer potential;
concept of an apparent size of the element ds as “seen” from the interior
point p and double layer potential
Maxwell-Lorentz equations, (retarded)
potentials and corresponding field
equations
Lorentz gauge
AA
c
because of the assumption
curlAH , 0)( c
AEcurl
and therefore
c
AE
Vector field A is not completely determined by the magnetic field H
Vector field A is completely determined by the magnetic field H ;
no differentiability assumption required; no calibration required due to this purely mathematical assumptions (w/o physical meaning; same situation
as for the “differential manifolds” assumptions for the Einstein field
equations)
[PlJ] Plemelj J., Potentialtheoretische Untersuchungen, B. G. Teubner Verlag, Leipzig, 1911 [ShF] Shu F. H., The Physics of Astrophysics, Volume I, Radiation, University Science Books, Mill Valley, California,1991
Appendix
The Eigenvalue problem for compact symmetric operators
In the following H denotes an (infinite dimensional) real Hilbert space with scalar product .,.
and the norm ... . We will consider mappings HHK : . Unless otherwise noticed the
standard assumptions on K are:
i) K is symmetric, i.e. for all Hyx , it holds KyxKyx ,,
ii) K is compact, i.e. for any (infinite) sequence nx bounded in H contains a
subsequence nx
such that nKx
is convergent,
iii) K is injective, i.e. 0Kx implies 0x .
A first consequence is Lemma: K is bounded, i.e.
x
KxK
x 0
sup:
.
Lemma: Let K be bounded, and fulfill condition i) from above, but not necessarily the two other
condition ii) and iii). Then K equals
x
KxxKN
x
,sup)(
0
.
Theorem: There exists a countable sequence ii ,
of eigenelements and eigenvalues
iiiK with the properties
i) the eigenelements are pair-wise orthogonal, i.e.
kiki ,,
ii) the eigenvalues tend to zero, i.e. i
i
lim
iii) the generalized Fourier sums
xxS i
n
i
in
1
,: with n for all Hx
iv) the Parseval equation
i
ixx22
,
holds for all Hx .
Hilbert Scales
Let H be a (infinite dimensional) Hilbert space with scalar product .,. , the norm ... and A be
a linear operator with the properties
i) A is self-adjoint, positive definite
ii) 1A is compact.
Without loss of generality, possible by multiplying A with a constant, we may assume
xAxx ,
for all )(ADx
The operator 1 AK has the properties of the previous section. Any eigen-element of K is also an eigen-element of A to the eigenvalues being the inverse of the first. Now by replacing
1 ii we have from the previous section
i) there is a countable sequence
ii , with
iiiA ,
kiki ,, and
ii
lim
ii) any Hx is represented by
(*) i
i
ixx
1
, and
1
22, ixx .
Lemma: Let )(ADx , then
(**) i
i
ii xAx
1
, ,
1
222,
i
ii xAx ,
ii
ii yxAyAx ,,,1
2
.
Because of (*) there is a one-to-one mapping I of H to the space H of infinite sequences of real numbers
,...),(ˆˆ:ˆ21 xxxxH
defined by
Ixx ˆ with ii xx , .
If we equip H with the norm
1
22,ˆ
ixx
then I is an isometry.
By looking at (**) it is reasonable to introduce for non-negative the weighted inner products
i
iiii
i
ii yxyxyx
,,ˆ,ˆ
and the norms
xxx ˆ,ˆˆ
2
Let H denote the set of all sequences with finite norm. then H is a Hilbert space. The
proof is the same as the standard one for the space 2l .
Similarly one can define the spacesH : they consist of those elements Hx such that
HIx ˆ with scalar product
i
iiii
i
ii yxyxyx ,,,
and norm
xxx ,
2 .
Because of the Parseval identity we have especially
yxyx ,,0
and because of (**) it holds
02
2, AxAxx ,
)(2 ADH .
The set 0H is called a Hilbert scale. The condition 0 is in our context necessary for
the following reasons: Since the eigen-values
i tend to infinity we would have for 0 : 0lim i. Then there exist
sequences ,...),(ˆ21 xxx with
2
2x ,
2
0x .
Because of Bessel’s inequality there exists no Hx with xIx ˆ . This difficulty could be
overcome by duality arguments which we omit here. There are certain relations between the spaces 0H
for different indices:
Lemma: Let . Then
xx
and the embedding HH is compact.
Lemma: Let . Then
xxx for
Hx
with
and
.
Lemma: Let . To any
Hx and 0t there is a )(xyy t according to
i)
xtyx
ii)
xyx ,
xy
iii)
xty )(
.
Corollary: Let . To any Hx and 0t there is a )(xyy t according to
i)
xtyx for
ii)
xty )( for .
Remark: Our construction of the Hilbert scale is based on the operator A with the two
properties i) and ii). The domain )(AD of A equipped with the norm
1
222,
i
ii xAx
turned out to be the space2H which is densely and compactly embedded in
0HH . It can be
shown that on the contrary to any such pair of Hilbert spaces there is an operator A with the properties i) and ii) such that
2)( HAD
0)( HAR and Axx
2.
Extensions and generalizations For 0t we introduce an additional inner product resp. norm by
1
2
)( ),)(,(),(i
ii
t
t yxeyx i
2
)(
2
)(),( tt
xxx .
Now the factor have exponential decay tie
instead of a polynomial decay in case of
i .
Obviously we have
xtcx
t),(
)( for
Hx
with ),( tc depending only from and 0t . Thus the normt )( is weaker than any norm .
On the other hand any negative norm, i.e.
x with 0 , is bounded by the norm0 and the
newly introduced normt )( . As it holds for any 0,, t and 1 the inequality
)(2 1
te one gets Lemma: Let 0 be fixed. The norm of any
0Hx is bounded by
2
)(
/2
0
22
t
t xexx
with 0 being arbitrary.
Remark: This inequality is in a certain sense the counterpart of the logarithmic convexity of the
norm , which can be reformulated in the form ( 0, , 1 )
2/22
xexx
applying Young’s inequality to
)()(222
xxx .
The counterpart of the fourth lemma above is
Lemma: Let 0, t be fixed. To any 0Hx there is a )(xyy t according to
i) xyx
ii) xy 1
1
iii) xeyx t
t
/
)(
.
In this paper we are especially concerned with the 2/1H Hilbert space, as the proposed
alternative framework to model quantum states. With respect to the above lemma this means, that for any bounded
0Hx it holds with :: t
1
212
0
2
)(
2
0
2
)(
/2
0
2
2/1i
it
t xexxexxexx i
.
For
0000 HH
with
100 , :
2
2/10
one therefore gets
1
212
0
2
2/1i
ixexx i .
Eigen-functions and Eigen-differentials
Let H be a (infinite dimensional) Hilbert space with inner product .,. , the norm ... and A be
a linear self-adjoint, positive definite operator, but we omit the additional assumption, that 1A
compact. Then the operator 1 AK does not fulfill the properties leading to a discrete spectrum. We define a set of projections operators onto closed subspaces of H in the following way:
),( HHLR
dE ,*)(:
0
, ,0 ,
i.e.
ddE ,*)( .
The spectrum CA )( of the operator A is the support of the spectral measure dE . The
set E fulfills the following properties:
i) E is a projection operator for all R
ii) for it follows EE i.e.
EEEEE
iii) 0lim
E and IdE
lim
iv)
EE
lim .
Proposition: Let E be a set of projection operators with the properties i)-iv) having a compact
support ba, . Let Rbaf ,: be a continuous function. Then there exists exactly one
Hermitian operator HHA f : with
),()(),( xxEdfxxA f .
Symbolically one writes
dEA .
Using the abbreviation
),(:)(, yxEyx ,
),(:)(, yxEdd yx one gets
)(),(),( , xxdyxEdyAx ,
)(,
22
1 xxdxEdx
)(),(),( ,
222 xxdyxEdyxA ,
)(,
2222 xxdxEdAx .
The function
2:)( xE is called the spectral function of A for the vector x . It has the
properties of a distribution function.
It hold the following eigen-pair relations
iiiA A 2
, )(),( .
The are not elements of the Hilbert space. The so-called eigen-differentials, which play a
key role in quantum mechanics, are built as superposition of such eigen-functions.
Let I be the interval covering the continuous spectrum of A . We note the following representations:
dxxxI
ii ),(),(1
,
dxxAxI
iii ),(),(1
dxxxI
i
2
1
22),(),(
,
dxxxI
ii
2
1
22
1),(),(
dxxAxxI
ii
222222
2),(),( .
Example: The location operator
xQ and the momentum operator
xP both have only a continuous
spectrum. For positive energies 0 the Schrödinger equation
)()( xxH
delivers no element of the Hilbert space H , but linear, bounded functional with an underlying
domain HM which is dense in H . Only if one builds wave packages out of )(x it results
into elements of H . The practical way to find Eigen-differentials is looking for solutions of a distribution equation.
Definition: Let with , i.e. is the boundary of the unit disk. Let being a
periodic function and denotes the integral from to in the Cauchy-sense. Then for
with and for real Fourier coefficients and norms are defined by
, .
The Fourier coefficients of the convolution operator
are given by
.
The operator enables characterizations of the Hilbert spaces and in the form
, .
This requires “differentiating / momentum building” of “less regular” “functions than .
It holds
, , ,
whereby
.
Lemma: It holds
i) if is an orthogonal system of , then is also an orthogonal
system of
ii)
iii) is an orthogonal system of
iv) for it holds
.
)(*
2 LH )( 21 RS )(su
2 0 2
)(: 2 LHu )(: 21 RS
dxexuu xi
)(2
1:
222
:
uu
dyyuyxkdyyuyx
xAu )()(:)(2
sin2log:))((
uukAu2
1)(
A 2/1H1H
0
2
2/12/1 ),( AH 0
2
11 ),( AAH
)(: 20 LH
01 ),(),( AvAuvu 02/1 ),(),( vAuvu 001 ),(),(),( HvHuvAuAvu
))(()(2
cot)(2
sin2log))(( xHudyyuyx
yudyx
xuA
n )1,0(#
2L nn H :
)1,0(#
2L
0,0nn
nn n )1,0(#
1H
#
0Hg
0001gHggAg
Some key properties of the Hilbert transform
are given in Lemma: i) The constant Fourier term vanishes, i.e.
ii)
iii)
For odd functions it hold
iv) If then and
are orthogonal, i.e.
v) ,
,
,
vi)
vii) If is an orthogonal system, so it is for the system ,
i.e.
viii) , i.e. if , then .
Proof:
ii) Consider the Hilbert transform of
.
The insertion of a new variable yields
iv) whereby is even .
dyyx
yuyd
yx
yuxHu
yx
)(1)(1lim:))((
0
0)( 0 Hu
dyyuxuxHxxuH )(1
))(())((
))(())(( xHuxxxuH
2, LHuu u Hu
0))()((
dyyHuyu
1H HH * IH 2 31 HH
gHfHgfgfH **)*( HgHfgf **
Nnn
NnnH )(
nnnnnn HHH ,,, 2
22uHu 2Lu 2LHu
)(xxu
dyyx
yyuxxuH
)(1))((
yxz
dyyuxuxHdzzxudzz
zxxudz
z
zxuzxxxuH )(
1))(()(
1)((1)()(1))((
dusigni
dyyHuyu2
)(ˆ)((2
))()((2
)(ˆ u
An ‘optimal order‘ shift theorem for the heat equation
For the parabolic model problem
fuu in T,0
0u on T,0
0)0( uu , in .
an ‘optimal order’ shift theorem in the Sobolev Hilbert space framework analogue to the Laplace equation is given in the form
Lemma: i)
T
k
T
kdtfcdtw
0
2
0
2
2
ii)
T
k
T
kdtftcdtwt
0
22/1
0
2
2
2/1
Proof: Let ),(: ii ww resp. ),(: ii ff being the generalized Fourier coefficient related to the
eigen pairs iii vv . Then it holds
)()()( tftwtw iiii and 0)0( iw .
with the solution
dfetw i
t
t
ii )()(
0
)(
.
By changing the order of integration one gets
T
i
t
t
t
tk
i
T
i
k
i
T
kdtdfededttwdtw ii
0
2
0
)(
0
)(2
0
22
0
2
2)()(
T T
t
i
k
i
T
i
t
t
i
k
i ddtefdtdfe ii
0
)(21
0
2
0
)(12 )()(
.
For the initial value problem it holds
0 zz in T,0)1,0(
0),1(),0( tztz for Tt ,0
)(),( xgoxz for )1,0(x .
The compatibility relations
0)1( g , 0)0( g , )1()1( 2gg , etc.
ensure corresponding regularity of the solution z In case the initial value function has reduced regularity the following estimates are valid
2)(2
)(l
lk
kgcttz , gcdtzt
T
0
2
2/1
2/1 .
An unusual ‘optimal order‘ shift theorem for the wave equation
We consider the hyperbolic model problem
fuu in T,0
0u on T,0
0)0( uu , 1)0( uu in .
An ‘optimal order’ shift theorem in the Sobolev Hilbert space framework analogue to the Laplace and the heat equation is not necessarily ensured for hyperbolic problems. This can be proven by a simple counter example:
Let 2))(
2
1(
:),(tx
etx
and ),(4),(2:),( txttxtxf
then, because of 0 and , it holds
Lemma: The function ),(:),( 2 txttxu is a solution of the wave equation with 010 uu
fulfilling
)()( 2222 LLLLu
while, at the same time, it holds
)()()( 222222 LLLLLLAuf .
The situation changed in case of the Hilbert space framework )( tH with the exponential decay
tie (and also for the related Hilbert scale
).( tH ):
Theorem: The hyperbolic self-adjoint, positive definite wave equation operator fulfills an ‘optimal order’ shift theorem in the form
)()( ).(2).(22 tktk HLHLAucu
.
Proof: The Fourier coefficients of the solution of the wave equation are given by
t
ii
i
i dfttu0
)()(sin1
)(
.
With this one gets
dtdfedteedtdftedttue
T t t
i
t
i
tt
i
T t
ii
t
i
T
i
t iiiii )))(())(sin(1
))()(sin(1
)(0 0 0
2)(2)(2
0 00
2
.
Applying the formula ([GrI] Gradshteyn/Ryzhik, table of integrals, 2.663)
)2sin()2cos(242
)(sin22
2 bxbbxa
ba
e
a
edxbxe
axaxax
this leads to (by changing the order of integration)
dfedtdefdtdfe i
T
i
T Tt
i
i
T t
i
t
i
iii )(1
)(1
)(1 2
0
2
0
2
2/3
0 0
2
2/3
.
Harmonic music “noise” signals “exactly” “between” red (Brownian) and white noise
Brownian noise or red noise is the kind of signal noise produced by Brownian motion. A Brownian motion (i.e. a Wiener process) is a continuous stationary stochastic process having independent
increments, i.e. )0()( BtB is a normal random variable with mean t and variance t2 , 2, constant
real numbers. The density function of a Brownian motion is given by
t
tx
tB et
xf2
2
2
)(
2)(
2
1)(
The sample paths of Brownian motion are not differentiable, a mathematical fact explaining the highly
irregular motions of small particles. The total variation of Brownian motion over a finite interval T,0 is
infinite. It holds
tt
tBVar
2)(
.
If )(tW is a Wiener process on the interval ,0 , then, as well the process
0
0
,
,
0
)/1(:)(*
t
tttWtW
.
White noise can be defined as the derivative of a Brownian motion (i.e. a Wiener process) in the framework of infinite dimensional distribution theory, as the derivative )(tB of )(tB , does not exist in the
ordinary sense. Not only )(tB , but also all derivatives of Brownian motion are generalized functions on
the same space. For each t , the white noise )(tB is defined as a generalized function (distribution) on
an infinite dimensional space. A Brownian motion is obtained as the integral of a white noise signal
)(tdB , i.e.
t
dBtB0
)()(
meaning that Brownian motion is the integral of the white noise )(tdB whose power spectral density is
flat
constBFourierS 2
0 )( .
This means, that the spectral density 0S for white noise is flat, i.e. cS 0
0 / i.e. it is inversely
proportional to 0 . It holds )()( BFourieriBFourier . Therefore the power spectrum of Brownian
noise is given by
2
02)()(
SBFourierS .
This means, that the spectral density of Brownian (red) noise is 2
0 /S , i.e. it is inversely proportional to
2 , meaning it has more energy at lower frequencies, even more so than pink noise.
The above indicates a spectral density for harmonic music “noise” signals given by
02
** )()(S
BFourierS .
Distribution solution of the 1D wave equation
The „vibration string“ equation
02 xxtt uku
has a solution )(),( ktxftxu for any function of one variable f , which has the physical
interpretation of a „traveling wave“ with „shape“ )(xf moving at velocity k .
There is no physical reason for the “shape” to be differentiable, but if it is not, the differential equation is not satisfied at some points. In order to not through away physically meaningful solutions because of technicalities, the concept of distributions can be applied. If the equation above is also meaningful, if u is a distribution, then u is called a weak solution of it. If u is twice continuously differentiable and the equation holds, one calls u a strong or classical solution. Each classical solution is a weak solution. In case of the equation above it’s also the other way around. The same is NOT TRUE for the elliptic Laplace equation (counter example is the classical solution )log(:).( 22 yxyxu , but not a weak solution as it holds there
4)log( 22 yx ). In order to see this we show that for )()(:).( 21 RLktxfyxu loc it holds
(*) 0),( 2 xxtt uku .
From the following identities
i) dxdttkxfuu tttttt )(),(),(
ii) dxdttkxfuu xxxxxx )(),(),(
it follows
dxdtktkxfuku xxttxxtt 22 )(),( .
Substituting the variable in the form ktxy and ktxz means
k
k
tx
zy
1
1
),(
),( and dydzkdxdt 2 .
From this it follows
dydzyfkdzdyyfkuku yzyzxxtt ))((2)(2),( 2 .
As
0
z
zyyzdz
this proves (*) above.
The time-symmetry of the classical wave equation
The time-symmetry of the classical wave equation is one of the main challenges of the big bang theory. It is linked to the miss-understanding that the ground state energy is fixed and uniquely defined: The definition of the Hamiltonian
VTxm
pH :
2
1
2: 22
2
_
defines the non-measurable ground state energy as the state of the lowest energy, i.e. at the point )0,0( px in the phase space, and defines this energy as zero.
The kinetic energy of strings with mass are given by
l
x dxtxuT0
2 ),(2
1 .
The internal forces of strings (being looked at as mechanical systems) are built on strains, depending proportionally from its lengths
l
x dxtxuL0
2 ),(1 .
For small displacements this is replaced by
l
x dxtxullL0
2 ...),(2
11 with
l
x dxtxul0
2 ),(2
1 .
Correspondingly the potential energy )(xV is approximately defined by
lLdL
dVllVllVLV )()()( .
Putting as the “strain constant”=”tension”
lLsdL
dV: and 0:)( lV
(the latter one in order to just simplify the algebraic term) the potential energy is then given in the form
l
xs dxtxuV0
2 ),(2
1 .
Defining the corresponding “string velocity” by
ssc :
then gives the wave equation of strings in the form
02 xxstt ucu .
Complementary variational principles
The topic is about „saddle point problems and non-linear minimization problems on convex
manifolds”. The scope is about saddle point problems and non-linear minimization problems on
sub Hilbert space 2/10 HHU .
Let RVVa :),( a symmetric bilinear form with energy norm
),(:2
uuau .
Let further Vu 0
and RVF :)( a functional with the following properties:
i) RVF :)( is convex on the linear manifold Uu 0
,
i.e. for every Uuvu 0, it holds )()()1())1(( vtFuFttvutF for every 1,0t
ii) )(uF for every Uuu 0
iii) RVF :)( is Gateaux differentiable, i.e. it exits a functional RVFu :)( with
)()()(
lim0
vFt
vFtvuFu
t
.
Then the minimum problem
min)(),(:)( uFuuauJ , Uuu 0.
is equivalent to the variational equation
0)(),( uFua for every U
and admits only an unique solution. In case the sub space U and therefore also the manifold Uu 0
is closed with respect to the
energy norm and the functional RVF :)( is continuous with respect to convergence in the
energy norm, then there exists a solution. We note that the energy functional is even strongly convex in whole V .
The method of Noble provides the link of the Hamiltonian function to the complementary variational principles:
Let )),(,( E and ),,( E be Hilbert spaces and EET : , EET :* linear operators fulfilling
uuTTuu ,, * .
Let further RxEEW : be a functional fulfilling
u
uWT
),( and u
uWT
),(*
i.e. the operators and derivatives from in the sense of Gateaux. Putting
)(),(2
1),( uFuuuuW
the minimization problem
(*) min)(2),(:)( uFTuTuuJ , EUu
leads to
uTu and )(),( * uFuT .
Lemma: If )(F is a convex functional it follows that ),( uuW is convex concerning u and
concave concerning u . Then the minimization problem (*) is equivalent to the variational
equation
0)(),( uFv for every U
resp.
)(),( * uFvT for every U .
In other words, there is a characterization of the solution of (*) as a saddle point.
References Arthurs A. M., Complementary Variational Principles, Clarendon Press, Oxford, 1970
Velte W., Direkte Methoden der Variationsrechnung, Teubner Studienbücher, B. G. Teubner Stuttgart,1976
Non Linear Problems and optimal FEM L error estimates
Let the problem be given by
0),( uxF
with the (roughly) regularity assumptions:
i) there is a unique solution
ii) uFF , are Lipschitz continuous.
The approximation problem is given by:
find hS 0)),,(( F for
hS .
Error analysis Put
))(,(:)( xuxFxf u and eu
Then
),(),( Rfe
with a remainder term
feeuFeRR ),(:)(:
resp.
)),((),( eRfufe .
Let hP denote the 2L projection related to ),(),( Rf , then
))(1
( eRf
uPh
resp.
)(:))(1
)( eTeRf
PuPIe hh .
Therefore the difference eue is a fix point of T .
For the following we use the abbreviations
L
eeB : and
LS
uh
inf: .
Key properties of the operator T
Lemma:
i) There is a 0 such that for sufficiently small, then T maps the ballB into itself.
ii) for sufficiently small, T is a contradiction in B .
Proof: i) Because of hP and 1f are being bounded it holds
LSLh ucPI
h
inf1
and
L
L
h eRceRf
P )())(1
( 2
.
It is 22
3
2
3),( cecfeeuFLL
with 3c being the Lipschitz constant of uF . Therefore
22
231)( ccceTL
.
Now fixing 1c and choosing
0 according to 0
2
231 ccc gives i)
ii) it holds
L
L
hLeReRceReR
fPeTeT )()())()((
1()()( 2122121
and
))((),((),(),()()( 212121 eeuFFeuFeuFeReR uu .
With ))1(,(),( 21 eeuFF uu
one gets
3),(),( cuFF uu .
Choosing )1
,(32
0
cc
Min
then proves ii).
Consequence: The operator T has a unique fix-point in the ball B
From this it follows the
Theorem: The FEM admits the error estimate
LSL
ucuh
inf .
With respect to “The regularity of piecewise defined functions with respect to scales of Sobolev spaces”
we refer to the corresponding paper of J. A. Nitsche.
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