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.