HQT: Notation, Glossary and Mathematical Definitions © 2005 Jochem Hauser and Walter Dröscher
14/05/2014 19:43
In the following some material for reference (notation,
glossary, and mathematical definitions) as
well as for review (mathematical definitions) is
given.
This material is constantly updated (August 4,
2005).
The notation in Appendix 1 describes the symbols
used in our publications concerning Heim
Quantum Theory (HQT).
The glossary of Appendix 2 describes the special
terminology used in HQT.
The glossary of Appendix 3 describes and explains
the special mathematical terminology used
in Heim's original work.
The mathematical definitions in Appendix 4 refer
to definitions used in modern physics, and are
meant to facilitate the reading of our papers.
Appendix 1: Notation and Physical
Constants
à value for the onset of conversion of photons
into gravitophotons.
A denotes the strength of the shielding potential
caused by virtual electrons.
Compton wave length of the electron
¤C= h
me c
=2.43×10−12m , ƛC=¤C /2¤.
c speed of light in vacuum 299,792,458 m/s ,
(1/c2 = e0 m0).
D diameter of the primeval universe, some
10125 m that contains our optical universe.
DO diameter of our optical universe, some
1026 m.
d diameter of the rotating torus.
dT vertical distance between magnetic coil and
rotating torus.
-e electron charge -1.602 × 10-19 C.
e¤z unit vector in z-direction.
Fe electrostatic force between 2 electrons.
Fg gravitational force between 2 electrons.
Fgp gravitophoton force, also termed Heim-Lorentz
force, Fgp=¤p e¤0 vT×H .
G = Gg + Ggp + Gq = 6.67259 × 10-11 m3 kg-1 s-2,
gravitational constant [1].
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HQT: Notation, Glossary and Mathematical Definitions
© 2005 Jochem Hauser and Walter Dröscher
Gg graviton constant, Gg≈G that is Gg decribes
the gravitational interaction without
the postulated gravitophoton and quintessence
interactions.
Ggp gravitophoton constant, Ggp≈¤1/67¤2Gg .
Gq quintessence constant, Gq≈4×10−18Gg .
gi k
¤ gp¤ metric subtensor for the gravitophoton
in subspace I2∪S2 (see glossary for subspace
description).
gi k
¤ ph¤ metric subtensor for the photon in subspace
I2∪S2∪T1 (see glossary for subspace
description).
h Planck constant 6.626076 × 10-34 Js,
ℏ=h/2¤.
hik metric components for an almost flat spacetime.
ℓ p= ¤G ℏ3
c3 =1.6×10−35m Planck length.
me electron mass 9.109390 × 10-31 kg.
m0 mass of proton or neutron 1.672623 ×
10-27 kg and 1.674929 × 10-27 kg.
Nn number of protons or neutrons in the universe.
q electric charge.
R distance from center of coil to location of
virtual electron in torus.
rN distance from nucleus to virtual electron in
torus.
R_ is a lower bound for gravitational structures,
comparable to the Schwarzschild radius.
The distance at which gravitation
changes sign, ρ, is some 46 Mparsec.
R+ denotes an upper bound for gravitation and
is some type of Hubble radius, but is not the
radius of the universe, instead it is the radius
of the optically observable universe. Gravitation
is zero beyond the two bounds, that is,
particles smaller than R- cannot generate
gravitational interactions.
re classical electron radius
re= 1
4¤¤0
e2
me c2=3 × 10−15m.
rge ratio of gravitational and electrostatic
forces between two electrons.
v velocity vector of charges flowing in the magnetic
coil, some 103 m/s in circumferential direction.
vT bulk velocity vector for rigid rotating ring
(torus) (see Sections. 3 and 4), some 103 m/s
in circumferential direction.
wgp probability amplitude (the square is the
coupling coefficient) for the gravitophoton
force (fifth fundamental interaction)
wgp
2 =Ggp
me
2
ℏ c
=3.87×10−49 probability amplitudes
(or coupling amplitudes) can be distance dependent.
wgpe probability amplitude for emitting a
gravitophoton by an electron
wgpe=wgp .
wgpa probability amplitude for absorption of a
gravitophoton by a proton or neutron
wgpa
2 =Ggpmp
me
ℏ c
.
wg_q conversion amplitude for the transformation
of gravitophotons and gravitons into the
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quintessence particle, corresponding to the
dark energy (rest mass of some 10-33 eV).
wph probability amplitude (the square is the
coupling coefficient for the electromagnetic
force, that is the fine structure constant α)
wph
2 = 1
4¤¤0
e2
ℏ c
= 1
137 .
wph_qp conversion amplitude for the transformation
of photons into gravitophotons.
wq probability amplitude for the quintessence
particle,(sixth fundamental interaction), corresponding
to dark energy (rest mass of
some 10-33 eV).
Z charge number (number of protons in a nucleus
of an atom).
Z0 impedance of free space,
Z0=¤¤0
¤0
≈376.7¤.
α coupling constant for the electromagnetic
force or fine structure constant 1/137.
αgp coupling constant for the gravitophoton
force.
γ ratio of probabilities for the electromagnetic
and the gravitophoton force
¤=¤wph
wgp¤2
=1.87×1046 .
¤0 permeability of vacuum 4π × 10-7 N/m2 .
t metron area (minimal surface 3Gh/8c3), current
value is 6.15¤10-70 m2.
Φ gravitational potential, Φ=GM/R.
ω rotation vector.
Abbreviations
BPP breakthrough propulsion physics
GP Geomtrization Principle
GR General Relativity
GRP General Relativity Principle
HQT Heim Quantum Theory
LQT Loop Quantum Theory
LHS left hand side
ls light second
ly light year
QED Quantum Electro-Dynamics
RHS right hand side
SR Special Relativity
VSL Varying Speed of Light
Subscripts
e electron
gp gravitophoton
gq from gravitons and gravitophotons into quintessence
ph denoting the photon or electrodynamics
sp space
Superscripts
em electromagnetic
gp gravitophoton
ph photon
T indicates the rotating ring (torus)
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Note: Since the discussion is on engineering
problems, SI units (Volt, Ampere, Tesla or
Weber/m2 ) are used. 1 T = 1 Wb/m2 = 104 G
= 104 Oe, where Gauss (applied to B, the
magnetic induction vector) and Oersted (applied
to H, magnetic field strength or magnetic
intensity vector) are identical. Gauss
and Oersted are used in the Gaussian system
of units. In the MKS system, B is measured
in Tesla, and H is measured in A/m
(1A/m = 4π × 10-3 G).
Note: For a conversion from CGS to SI units,
the electric charge and magnetic field are replaced
as follows:
e¤e/¤4¤¤0 and H ¤¤4¤¤0H .
Appendix 2: Glossary of Physical
Terms
aeon Denoting an indefinitely long period of
time. The aeonic dimension can be interpreted
as steering structure governed by the
entelechial dimension toward a dynamically
stable state.
apeiron the unlimited primeval substance in
Greek natural philosophy Used to characterize
the state of existence before the quantized
bang, similar to Penrose's mathematical
world or world of potentialities.
anti-hermetry Coordinates are called anti-hermetric
if they do not deviate from Cartesian
coordinates, i.e., in a space with anti-hermetric
coordinates no physical events can
take place.
canonical Conforming to a general rule or acceptable
procedure, a canonical form is the
simplest form possible (for instance a unit
matrix).
condensation For matter to exist, as we are used
to conceive it, a distortion from Euclidean
metric or condensation, a term introduced by
Heim, is a necessary but not a sufficient condition.
condensor The Christoffel symbols of the second
kind ¤k m
i become the so called condensor
functions, ¤i
km that are normalizable.
It can be shown that ¤i
km the have tensor
character. It should be noted that in the eigenvalue
equations for the mass spectrum of
elementary particles, the ¤i
km are eigenfunctions
and thus must not be confused with the
Christoffel symbols. It should be mentioned
that Heim first writes a symbolic eigenvalue
equation that he later on using symmetry arguments
transforms into a mathematically
correct eigenvalue equation. The term condensor
is derived from the fact that these
functions represent condensations of the
spacetime metric. The condenser is an opera-
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tor projecting a deformation in the 6, 8, or
12-dimensional metronic lattice into ℝ4,
where it appears as an intricate, geometrically
structured, compressed or “condensed”
lattice configuration. This condensed, structured
region is what eventually is interpreted
as matter constituting an elementary particle.
condensor flux
conjunctor
conversion amplitude Allowing the transmutation
of photons into gravitophotons, wph_gp
(electromagnetic-gravitational interaction),
and the conversion of gravitophotons into
quintessence particles, wgp_q (gravitationalgravitational
interaction).
coupling constant Value for creation and destruction
of messenger (virtual) particles,
relative to the strong force (whose value is
set to 1 in relation to the other coupling constants).
coupling potential between photon-gravitophoton
(Kopplungspotential) As coupling
potential the term gi k
¤gp¤ of the metric is denoted.
The reason for using the superscript
gp is that this part of the photon metric
equals the metric for the gravitophoton particle
and that a →sieve (conversion) operator
exists, which can transform a photon into a
gravitophoton by making the second term in
the metric anti-hermetric. In other words, the
electromagnetic force can be transformed
into a repulsive gravitational like force, and
thus can be used to accelerate a material
body.
cosmogony (Kosmogonie) The creation or origin
of the world or universe, a theory of the
origin of the universe (derived from the two
Greek words kosmos (harmonious universe)
and gonos (offspring)).
covariant For different inertial frames the laws
of physics are varying so as to preserve the
mathematical form of these laws. For instance,
Newton's law of gravitation is not
covariant under a →Lorentz transformation.
entelechy (Greek entelécheia, objective, completion)
used by Aristotle in his work The
Physics. Aristotle assumed that each phenomenon
in nature contained an intrinsic objective,
governing the actualization of a
form-giving cause. The entelechial dimension
can be interpreted as a measure of the quality
of time varying organizational structures (inverse
to entropy, e.g., plant growth) while
the aeonic dimension is steering these structures
toward a dynamically stable state. Any
coordinates outside spacetime can be considered
as steering coordinates.
differentiable manifold Contains a collection of
points, each of which determines a unique
position in →Heim space. The smoothness
feature is only applicable in the case where
the physical problem considered involves a
large number of →metrons. Continuous and
differentiable functions are supported. The
differentiable manifold is a topological space
(open sets) and is locally equivalent to a
space ℝn which is of the same dimension as
the corresponding Heim space, i.e., there exists
a one-to-one mapping between the open
sets of the Heim space and the ℝn.
energy coordinates
epistemology Theory of the nature of knowledge
especially with reference to its limits
and validity.
eschatology Concerned with the final events in
the history of the universe.
event
field activator (Feldkaktivator) flips the spin of
a metron, i.e., changes its orientation.
flucton (Flukton) being movable and compressible.
flux aggregate
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fundamental kernel (Fundamentalkern) Since
the function ¤im
¤¤¤ occurs in xm
¤¤¤=∫¤im
¤¤¤d ¤i
as the kernel in the integral, it is denoted as
fundamental kernel of the poly-metric.
Galilean spacetime A spacetime in which the →
Galilei transformation is valid
Galilei transformation
geodesic zero-line process This is a process
where the square of the length element in a
6- or 8-dimensional →Heim space is zero.
gravitational limit(s) There are three distances
at which the gravitational force is zero. First,
at any distance smaller than R_, the gravitational
force is 0. Second,
gravitophoton (field) Denotes a gravitational
like field, represented by the metric sub-tensor,
gi k
¤ gp ¤ , generated by a neutral mass with
a smaller coupling constant than the one for
gravitons, but allowing for the possibility
that photons are transformed into gravitophotons.
Gravitophoton particles can be
both attractive and repulsive and are always
generated in pairs from the vacuum under
the presence of virtual electrons. The total
enery extracted from the vacuum is zero, but
only attractive gravitophotons are absorbed
by protons or neutrons. The gravitophoton
field represents the fifth fundamental interaction.
The gravitophoton field generated by
repulsive gravitophotons, together with the
→vacuum particle, can be used to reduce the
gravitational potential around a spacecraft.
graviton (Graviton) The virtual particle responsible
for gravitational interaction.
heimline In analogy to →worldline, the path of
a particle in →Heim space.
Heim-Lorentz force Resulting from the newly
predicted gravitophoton particle that is a
consequence of the →Heim space H8. A metric
subtensor is constructed in the subspace
of coordinates I2, S2 and T1, denoted as hermetry
form H5. The equation describing the
Heim-Lorentz force has a form similar to the
electromagnetic Lorentz force, except, that
it exercises a force on a moving body of
mass m, while the Lorentz force acts upon
moving charged particles only. In other
words, there seems to exist a direct coupling
between matter and electromagnetism. In
that respect, matter can be considered playing
the role of charge in the Heim-Lorentz
equation. The force is given by
Fgp=¤p e¤0 vT×H . Here Λp is a coefficient,
vT the velocity of a rotating body (insulator)
of mass m, and H is the magnetic field
strength. It should be noted that the gravitophoton
force is 0, if velocity and magnetic
field strength are parallel.
Heimian metric
Heim space A Heim space is a discrete space of
6, 8, or 12 dimensions, denoted as H6, H8,H12
respectively, with three spatial coordinates
of + signature, while any other coordinate
has - signature. In a Heim space an elemental
surface, termed →metron, exists. If the
surface considered comprises a large number
of metrons, a Heim space can be considered
a →differentiable manifold endowed with a
→Heimian metric.
hermetry form (Hermetrieform) The word
hermetry is an abbreviation of hermeneutics,
in our case the semantic interpretation of the
metric. To explain the concept of a hermetry
form, the space H6
is considered. There are 3
coordinate groups in this space, namely
s3=¤¤1 ,¤2 ,¤3¤ forming the physical space
ℝ3, s2=¤¤4¤ for space T1, and
s1=¤¤5 ,¤6¤ for space S2. The set of all
possible coordinate groups is denoted by S=
{s1, s2, s3}. These 3 groups may be combined,
but, as a general rule (stated here
without proof, derived, however, by Heim
from conservation laws in H6, (see p. 193 in
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Elementarstrukturen der Materie, Vol I,
Resch Verlag 1996), coordinates ¤5 and ¤6
must always be curvilinear, and must be present
in all metric combinations. An allowable
combination of coordinate groups is termed
hermetry form, responsible for a physical
field or interaction particle, and denoted by
H. H is sometimes annotated with an index,
or sometimes written in the form H=(¤1, ¤2
,...). This is a symbolic notation only, and
should not be confused with the notation of
an n-tuple. From the above it is clear that
only 4 hermetry forms are possible in H6. It
needs a →Heim space H8 to incorporate all
known physical interactions. Hermetry
means that only those coordinates occurring
in the hermetry form are curvilinear, all other
coordinates remain Cartesian. In other
words, H denotes the subspace in which
physical events can take place, since these
coordinates are non-euclidean. This concept
is at the heart of Heim's geometrization of all
physical interactions, and serves as the correspondence
principle between geometry
and physics.
hermeneutics (Hermeneutik) The study of the
methodological principles of interpreting the
metric tensor and the eigenvalue vector of
the subspaces. This semantic interpretation
of geometrical structure is called hermeneutics
(from the Greek word to interpret).
hermitian matrix (self adjoint, selbstadjungiert)
A square matrix having the property
that each pair of elements in the i-th row
and j-th column and in the j-th row and i-th
column are conjugate complex numbers (i ¤
- i). Let A denote a square matrix and A* denoting
the complex conjugate matrix. A† :=
(A*)T = A for a hermitian matrix. A hermitian
matrix has real eigenvalues. If A is real, the
hermitian requirement is replaced by a requirement
of symmetry, i.e., the transposed
matrix AT = A .
homogeneous The universe is everywhere uniform
and isotropic or, in other words, is of
uniform structure or composition throughout.
hyperstructure (Hyperstruktur) Any lattice of
a →Heim space that deviates from the isotropic
Cartesian lattice, indicating an empty
world, and thus allows for physical events
to happen.
inertial transformation (Trägheitstransformation)
Such a transformation, fundamentally
an interaction between electromagnetism and
the gravitational like gravitophoton field, reduces
the inertial mass of a material object
using electromagnetic radiation at specific
frequencies. As a result of momentum and
energy conservation in 4-dimensional spacetime,
v/c = v'/c', the Lorentz matrix remains
unchanged. It follows that c < c' and v < v'
where v and v' denote the velocities of the
test body before and after the inertial transformation,
and c and c' denote the speeds of
light, respectively. In other words, since c is
the vacuum speed of light, an inertial transformation
allows for superluminal speeds.
An inertial transformation is possible only in
a 8-dimensional →Heim space, and is in accordance
with the laws of SRT. In an Einsteinian
universe that is 4-dimensional and
contains only gravitation, this transformation
does not exist.
isotropic The universe is the same in all directions,
for instance, as velocity of light transmission
is concerned measuring the same
values along axes in all directions.
Lorentzian metric
Lorentz transformation This transformation in
spacetime reflects the fundamental fact that
light travels with exactly the same speed c
with respect to any inertial frame
x'= x−vt
¤1−v2/c2¤1/2 , y'=y , z '=z ,
t'= t−vx /c2
¤1−v2 /c2¤1/2 .
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Minkowski spacetime A spacetime in which the
→Lorentz transformation is valid.
ontology A particular theory about the nature of
being.
partial structure (Partialstruktur) For instance,
in H6, the metric tensor that is hermitian
comprises three non-hermitian metrics
from subspaces of H6. These metrics from
subspaces are termed partial structure.
poly-metric The term poly-metric is used with
respect to the composite nature of the metric
tensor in 8D →Heim space. In addition, there
is the twofold mapping ℝ4® H8® ℝ4.
probability amplitude With respect to the six
fundamental interactions predicted from the
→poly-metric of the →Heim space H8, there
exist six (running) coupling constants. In the
particle picture, the first three describe gravitational
interactions, namely wg (graviton, attractive),
wgp (gravitophoton, attractive and
repulsive), wq (quintessence, repulsive). The
other three describe the well known interactions,
namely wph (photons), ww (vector bosons,
weak interaction), and ws (gluons,
strong interaction). In addition, there are
two →conversion amplitudes predicted that
allow the transmutation of photons into
gravitophotons (electromagnetic-gravitational
interaction), and the conversion of
gravitophotons into quintessence particles
(gravitational-gravitational interaction).
protosimplex
prototrope (Prototrope) first in time *protohistory*
b : beginning : giving rise to *protoplanet
quantized bang According to Heim, the universe
did not evolve from a hot big bang, but
instead, spacetime was discretized from the
very beginning, and such no infinitely small
or infinitely dense space existed. Instead,
when the size of a single metron covered the
whole (spherical volume) universe, this was
considered the beginning of this physical universe.
That condition can be considered as
the mathematical initial condition and, when
inserted into Heim's equation for the evolution
of the universe, does result in the initial
diameter of the original universe [1]. Much
later, when the metron size had decreased far
enough, did matter come into existence as a
purely geometrical phenomenon.
selector (Selektor)
shielding field (Schirmfeld)
sieve operator see → transformtion operator
transformation operator or sieve operator
(Sieboperator) The direct translation of
Heim's terminology would be sieve-selector.
A transformation operator, however, converts
a photon into a gravitophoton by making
the coordinate ¤4 Euclidean.
vacuum particle responsible for the acceleration
of the universe, also termed quintessence
particle The vacuum particle represents
the sixth fundamental interaction, and
is a repulsive gravitational force whose
gravitational coupling constant is given by
4.3565×10-18 G. It only interacts with gravitons
and positive (repulsive) gravitophotons,
but not with real or virtual particles.
worldline Path of a particle in spacetime ℝ4.
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Appendix 3: Heim's Original
Mathematical Terminology
Appendix 4: Mathematical Definitions
affine transformation is defined by a mapping
(→lambda matrix) from coordinates x ¤ x':
x¤'=¤¤
¤ x¤¤a¤ that guarantees the invariance
of the spacetime interval
¤ xB – xA¤2−c2¤t B−t A¤2=0 between two
→worldpoints A and B. It is necessary that
the vacuum speed of light as an upper limit
remains invariant between any two inertial
systems. The →Lorentz transformation satisfies
this requirement.
bijective mapping A mapping f is bijective if f
is both →injective and →surjective. The
→inverse mapping f-1 is defined by f x0¤
x0 that is, f-1 associates with each y0∈Y the
corresponding x0∈D(f) for which f x0 = y0.
boost parameter
Casimir invariants the scalar product of →Lie
group generators is known as Casimir invariant
that commutes with all the generators
(e.g., J 2 commutes with J1, J2, and J3)
and is therefore invariant under all group
transformations. The eigenvalues of the Casimir
invariants are the conserved quantum
numbers of the symmetry group. The groups
O(3) and SU(2) have only one invariant
while the SU(3) group has two invariants.
Clifford algebra in the Dirac equation four constant
coefficients ¤¤ occur that are non-commutable
square matrices. Thus the wavefunction
ψ(x) is a column matrix.
The ¤ matrices satisfy the condition
{¤¤¤¤ }:=¤¤¤¤¤¤¤¤¤=2¤¤¤ where ¤¤¤
is the Minkowski space-time metric tensor
(diagonal form). The ¤ matrices are 4×4
matrices.
cotangent space consider a point P and its
→tangent space TPM on a →manifold M. Let
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xμ(λ) be a smooth →curve through P on M.
The directional derivative of a smooth function
f on M at P (tangential vector v) with
respect to the curve xμ(λ) is a differentiable
mapping TPM¤ℝ. The set of all →oneforms
ω(v) at P that map TPM¤ℝ forms the
dual vector space to TPM, termed the cotangent
space T*
PM. Bundling together all T*
PM
at different points on M, gives the cotangent
bundle T*M.
curve in a manifold M if ¤ is chosen to be the
distance along the curve, a curve parametrized
by ¤ has the tangent vector
¤dx¤/d ¤¤ ,¤=1,2,. .. , n.
diffeomorphism A differentiable, bijective mapping
f for which both f and f-1 are smooth,
i.e., arbitrarily often differentiable (note: a
homomorphism only requires that f and f-1
are continuous). It should be noted that a
diffeomorphic mapping of generalized coordinates
q to coordinates q' leaves the equations
of motions invariant that are derived
from Lagrange function L.
differentiable manifold A →manifold can be
covered by patches (charts). Different coordinate
systems can be set up on any part of a
manifold. For two overlapping patches, two
different coordinate systems can be defined
in the overlap region, xμ and xμ'= xμ'(xμ ).
Any function f in the overlap region that is
differentiable with respect to coordinates xμ
should also be differentiable with regard to
xμ'. This is true if the transformation xμ'(xμ )
between the two coordinate systems is differentiable.
Such a manifold is called a differentiable
manifold. If there exist n derivatives
there is a Cn manifold. For n=∞, the manifold
is C∞.
dual space →cotangent space.
fiber bundle
groups:
compact Lie group a compact Lie group is a
→Lie group whose parameters are defined in
a closed interval, for instance, for U(1) angle
θ varies in the interval [0, 2π].
continuous groups contain an infinite number
of elements. A simple example is the set of
all complex wavefactors of a wavefunction
in quantum mechanics written in the form U
(θ)=e iθ . The product of two phasefactors is
U(θ) U(θ')= U(θ+θ') and the inverse is given
by U-1(θ)=U(-θ). These phase factors from a
group called U(1). This group is characterized
by a single parameter, the angle θ in the
interval [0, 2π]. The group is differentable
since dU = i U dθ and thus the derivative is
an element of the U(1). The group of rotations
in three-dimensional space the →O(3)
group and the group of →Lorentz transformations
are also continuous groups.
Lie group the characteristic of a Lie group is
that the parameters of a product are analytic
functions of each parameter in the product
that is, if U(θ) =U(α) U(β) then θ=f(α, β)
where f is an analytic function. The analytic
property guarantees that the group is differentiable
so that an infinitesimal group element
dU(θ) can be defined. The →Lorentz
group is an example of a noncompact group.
The boosts or transformations from one inertial
frame to another are represented by nonunitary
matrices. The →boost parameter
η=tanh-1(v/c) is not restricted to a closed interval.
Lorentz group
orthogonal groups O(n) is the group of rotations
in an n-dimensional Euclidean space.
The elements of O(n) are n×n real valued
matrices that have n(n-1)/2 independent parameters.
O(3) is the three-dimensional rotation group
and leaves the distance x2+y2+z2 invariant.
The parameters are the Euler angles α, β, γ
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from classical mechanics. The rotation matrix
is denoted by R(α, β, γ) and can be written
as a sequence of three rotations.
Poincaré group
special unitary groups SU(n) has
det SU(n)=+1 with (n2-1) independent elements.
special unitary group SU(2) and SU(3) represent
two- and three-dimensional matrices
and are associated with isotopic spin and
color. A SU(2) rotation leaves the magnitude
of the original vector invariant and has
det SU(2)=1.
The SU(3) group has eight independent
→group generators, represented by 3×3
→hermitian matrices and denoted as Fi,
i=1,...,8. The matrices obey special commutation
rules.
unitary group the elements of the unitary group
U(n) are given by n×n →unitary matrices.
The determinant det U(n)=¤1.
group generator:
Lie group generator let G be a Lie group and
operators Fk be its group generator in analogy
to the angular momentum operator Jk
for the →rotation group. The operator Fk
generates an element of G in the same way
that Jk generates a rotation, i.e.,
U=e−i¤k Fk . The number of generators is
equal to the number of independent parameters
in G. Thus there are n(n-1)/2 generators
for O(n) and n2-1 generators for SU(n). Generators
of the orthogonal and unitary groups
are represented by →hermitian or self-adjoint
matrices. The commutation rule is
given by [Fi, Fk]=i cikm Fm. The cikm are called
the structure constants. For O(3) these values
are either ±1 or 0. The structure of the is
different from the group structure. The form
the basis of a linear vector space that is
known as Lie algebra. There exists both a
scalar product and a vector product (in form
of the commutation relation). This vector
product is also called Lie product. For instance,
for the total angular momentum
J2=J1
2 +J2
2 +J3
2 .
rotation group the rotation of wavefunction
about a direction given by unit vector n is
written as R¤¤¤=e−i ¤ ¤n⋅J , where J is the
angular momentum operator. For an angle
dθ, the rotation is given in first order as
R¤¤¤=1−id¤¤n⋅J . The combined rotations
about the x and y-axis is given by
R¤¤¤=¤1−i d ¤ J 1¤¤1−i d ¤ J 2¤. Reversing
the order of rotations and forming the
difference is written as the commutator
[J1, J2] dθ dϕ. Angular momentum operators
obey the commutation relation
[Ji, Jk]=i εikm Jm. Operators Jm are called the
group generators. This means that O(3) belongs
to a non-commutative or non-Abelian
group. In contrast, O(2) and U(1) are Abelian
groups.
hermitian matrix
homeomorphism is a →bijective mapping
f: A¤B for which both f and f-1 are continuous.
homomorphism of O(3) and SU(2) there is a
homomorphism (many-to-one mapping) between
O(3) and SU(2) that is, a real three-dimensional
rotation can be associated with an
element of SU(2). In other words, a vector
(x, y, z) can be defined from the complex
vector being transformed by SU(2) which is
left invariant. It is possible to associate the
general matrix R(α, β, γ) with the complex
parameters of an SU(2) matrix. This means
that the 3×3 real valued matrix O(3) can be
expressed by the 2×2 complex valued matrix
SU(2). However, the relation between the
two matrices is not one-to-one, hence the
denotation homomorphism ( here two possibilities
exist).
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injective mapping a mapping is injective if for
every x1≠x2∈D(f), x1≠x2 implies f x1≠ f x2
that is, different points in the domain have
different images. Therefore, the inverse image
is a single point in D(f).
inverse mapping for an →injective mapping
f : D(f)¤ Y the inverse mapping f-1 is defined
as to be the mapping R(f) ¤ D(f) such that
y0 ∈R(f) is mapped onto that x0∈D(f) for
which f x0= y0.
lambda matrix given are two coordinate systems
x and x´ on a manifold. The transformation
matrix Λ, →spinor, is defined as
¤¤
¤´=∂ x¤´
∂ x¤ .
Lie algebra →group generator, see Lie group
generator.
Lorentz transformation this transformation
(→affine transformation) in spacetime reflects
the fundamental fact that light travels
with exactly the same speed c with respect
to any inertial frame (here x and x' denote
coordinates)
x'= x−vt
¤1−v2/c2¤1/2 , y'=y , z '=z ,
t'= t−vx /c2
¤1−v2/c2¤1/2 .
manifold A manifold is locally equivalent to an
n-dimensional euclidean space ℝn. That is
manifold M has the same local topology. M
therefore must be a topological space, which
means there is a collection of open sets that
cover it. Second, the structure of these open
sets, within small regions, is equivalent to
the natural topology of ℝn. ℝn is a Haussdorf
space, which means that for any two different
points there exist non-overlapping neighborhoods,
and there exists a basis B in form
of a collection of open sets such that each
subset of ℝn can be represented by a union
of elements of B. Therefore for manifold M
it is required that every point of M belongs
to at least one open set of its basis B and
there exists a one-to-one correspondence
with the points of some open set of ℝn . This
means that there is a continuous →bijective
mapping from the open set of M to the open
set in ℝn. When these conditions are met, M
is called a manifold. Any function f(P) for
each point P¤M can be re-expressed as a
function g(xμ) defined on ℝn. A manifold
does not possess a metric, hence no scalar
product can be defined on a manifold, →oneform.
mapping Given two sets X and Y with A⊂X. A
mapping f from A into Y associates with each
x∈X a single y∈Y, called the image of x with
respect to f, and written as y=f x. The set A
is called the domain, D(f), of f. The range of
f, R(f), is the set of all images. The inverse
image of element y0∈Y is the set of all
x∈D(f) such that f x= y0.
one-form one-forms are the extension of the
scalar product of u v of two Euclidean vectors
to a →manifold M. On M no metric is
defined, therefore a salar product cannot be
formed. A scalar product is considered as a
function. For a given vector u, the symbol u•
acts as a function that assigns to each vector
v a real number. This mapping is linear. A
one-form ω on M is therefore defined as a
linear mapping that is real-valued. Because
of the linearity ω(v)=ωμvμ. Indices of a oneform
are in the lower position. A one-form
field is defined in the same way as a linear
function on vector fields. An example for a
one-form field is the gradient of a scalar field
f, denoted as ¤¤ f. Taking as v the tangent
vector d/dλ to a curve xμ(λ), one can write
ωf ¤ v¤= ∂ f
∂ x¤
∂ x¤
∂¤
.
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partial derivative ¤¤:=¤/¤x¤. The partial derivatives
are sometimes also considered as base
vectors in the corresponding coordinate systems,
see →one-form.
rank of tensor field tensors and tensor fields are
classified by their rank, denoted as ¤j
i¤.
Rank ¤00
¤ are called scalars and are real
numbers. A scalar field is a real valued function
f(P) on the manifold. Rank ¤10
¤ are
called contravariant vectors. They correspond
to a tangential vector on a curve in
differential geometry. In a coordinate system
a vector can be resolved in its components.
In general the upper variable denotes the
number of contravariant indices and the
lower one the covariant indices of a tensor.
spinor the wavefunction of the Dirac function is
a 4-component column matrix. These components
will be expressed by a different set
of functions and also be rearranged when
transformed from coordinate system x to x.
Since the components of are not components
of a spacetime vector, but represent a
state in which a particle can exist, they do
not transform as vector components, i.e.,
they do not follow any tensor transformation
law The transformation law for the ψβ components
is given by ¤¤
' =S¤¤¤¤¤¤¤ . The
matrix S is determined from the covariant
form of the Dirac equation under a Lorentz
transformation. For matrix Λ see →lambda
matrix.
structure constants cijk →Lie group generator.
surjective mapping
tangent space consider a point P on a manifold
M, for instance a point on a sphere embedded
in ℝ3. The set of all tangent vectors at P
of all curves through P forms a vectorspace
over ℝ, denoted by TPM as the tangent space
to M at P. All tangent vectors are in the tangential
plane through P. For the example of
the sphere TPM is the vetorspace ℝ2.
tangent bundle The disjoint union of all →tangent
spaces TM :=∪P
T PM is itself a
manifold of dimension 2n if M has dimension
n. The individual tangentspace TPM at point
P is called a fiber.
tensor field assigns a physical property to every
point on the manifold.
unitary matrix (unitär) let A denote a square
matrix, and A* denoting the complex conjugate
matrix. If A† := (A*)T = A-1, then A is a
unitary matrix, representing the generalization
of the concept of orthogonal matrix. If
A is real, the unitary requirement is replaced
by a requirement of orthogonality, i.e.,
A-1 = AT. The product of two unitary matrices
is unitary.
unitary transformation used in quantum mechanics,
leaving the modulus squared of the
a complex wavefunction invariant, → unitary
group.
world point, event, worldline a point (x, t) that
specifies both the spatial coordinates and
time is called a world point or event. The
evolution of a signal, represented by a parametrized
curve (x(t), t) is termed worldline.
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[1] Woan, G., The Cambridge Handbook of
Physics Formulas, Cambridge University Press,
2000.,
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