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Space, Mass, and Quarks
P. Żenczykowski
Institute of Nuclear Physics
Polish Academy of Sciences
Kraków, Poland
May 11, 2006
Physics provides
DESCRIPTION of phenomena
In order to describe phenomena we use language
Over centuries we have found that a very useful description
is provided by the language of mathematics
Consequently, some people think
that mathematics may lead us in our attempts to describe Nature
This is erroneous
Mathematics does not lead anywhere
It just provides a tool which facilitates
and quantifies the description and the idea behind it
In reality we are almost totally blind, we have no maps
of new territory - it was, it is, it always will be like that
Perfect figure
Pythagoras - the most perfect figure is a sphere
Vision,
intuition,
aesthetics
r2
2
F  m1 m
2
r
2
F  m1 m
2
r
Spherically
symmetric
r2 - SO(3) scalar
Macroscopic classical space and elementary particles
At least some of the attributes of observed particles are closely
connected with the properties of macroscopic space
Examples:
Spin – rotation
Parity, chirality – space reflection P
Existence of particles and antiparticles – C – CPT
– (Antiparticles as particles moving backward in time)
Other quantum numbers – internal symmetries? Flavour, colour?
There were attempts – no connection seen so far
NO GO theorems
However: such NO GO theorems are valid only EXACTLY in the form they were proven
In fact: there exists an enormous space of untried concepts, ways, and possibilites
HYPOTHESIS:
All attributes of elementary particles are connected
with the properties of macroscopic classical arena
on which physical processes are deemed to occur
Roger Penrose:
(„Structure of spacetime”, 1968)
I do not believe that a real understanding of
the nature of elementary particles can ever
be achieved without a simultaneous deeper
understanding of the nature of spacetime
Other physicists of similar opinions:
Wheeler: „How could one believe any account of the foundation for the central structure
of physics, spacetime, which proceeded without reference to the quantum,..?”
Finkelstein: „inherent spacetime illusion”
Einstein: „Time and space are modes by which we think, …”
Nonrelativistic or relativistic description ?
Spin – rotation - nonrelativistic
Parity – reflection - nonrelativistic
For example one may linearize
Schrödinger equation 2mE=p2
Existence of left and right
does not have much to do
with Lorentz transformations
and different transformations of
 L and  R
Invariance under reflections requires that
p2  p    1
 = 0  3
p  –p
Somebody may say:
but relativistic Dirac equation
not only yields
  1     1 –   1proper gyromagnetic ratio g=2
[e (L + 2 S)B/m]
but also leads to ANTIPARTICLES
Left and Right, Dirac matrices -
- no relativity needed
Gyromagnetic:
Levy-Leblond 1971
This is true.
But it does NOT follow from relativity!
Particles to antiparticles?
Nonrelativistic description is enough: (Horzela, Kapuścik; P.Ż)
In words: no relativity is needed in the idea of antiparticles
as particles moving „backwards in time”
In formulas: in linearized Schrödinger equation we must obtain the term
(p – e A)    1  (p – e A)  
As for Dirac case: complex conjugation yields
But p* = – p, and C * C-1 = 
( C = – i 2  2 )
„e”
(p* – e A)  *
hence – (p + e A)  
change of relative sign at
This suggests that NONRELATIVISTIC description
should be sufficient (at least at the beginning) when
trying to extend our understanding of quantum numbers
of elementary particles in terms of space concepts
Mass
Higgs mechanism:
(+) renormalizability with massive gauge bosons
() THEORY: just shifting the problem
constant term in energy density of vacuum
55 orders of magnitude too large
EXPERIMENT: not observed (?)
PHILOSOPHY: tendency to „explain” phenomena in
terms of material objects rather than
abstract concepts („heat fluid”, …)
Particle and quark masses
Free particles mp, mn, m
F=evxB
F = ma

THEORETICAL
CALCULATIONS
LQCD(mq)
mp= fp,theory(mq,…)
e/m, m
theory used to extract mass
i.e. again: theory used to extract mass
Conceptually identical
method actually used
PROBLEM
method NOT used
Constituent, current quark masses
constituent – current with interaction
A  q1  5q2
2
m     A  q1 p
1  p
 2  5q2  m1  m2q1 5q2
external quarks
current:
2
2
m , mK  mu
md , md ms
Nothing in common with
mp = fp,theory(mq,…)
Constituent – e.g. magnetic moments
Free Dirac quarks again
Free Dirac equation used
Quarks on mass shell
Plane, infinite waves
No confinement
CONCEPTUAL
NONSENSE
Quark mass – internal quarks
Trees
Loops
WRHD
s
u
u
charmed quark mass
s
d
d
s
Gaillard, Lee (1974)
 1.5-2.0 GeV
No quark model pole
Quark mass through propagator
No: Zweig
Yes: Gell-Mann SU(3)L SU(3)R
meson formfactors
(Schwinger – magnetic moments)
DEEPLY
1) Conceptual problems & internal inconsistencies
DISSATISFYING: 2) Point 1 either unnoticed or „swept under the carpet”
Mass problem
Quark mass - no standard propagators (trees)
- some meaning in loops?
- extension of concept ?
 detach concept of mass from concept of standard propagator
 broader structure needed
One basic constant needed: e.g. m0
– other masses from (unknown) theory through dimensional mass ratios
If (c), ħ added – the basic constant may be of dimension GeV/cm
( Fundamental length, momentum)
M. Born -1949 – „Reciprocity theory of elementary particles”
„I think that the assumption of the observability of the 4-dimensional
distance of two events inside atomic dimensions (no clocks or
measuring rods) is an extrapolation…
…I am inclined to interpret the difficulties which QM encounters in
describing elementary particles and their interactions as indicating
the failure of that assumption
There is of course a quantity analogous to R=t2-x2, namely P=E2-p2 ,
not continuous (square of rest mass). A determination of P is not a real
measurement, but a choice between a number of values corresponding
to the particles…
It looks, therefore, as if the distance P in momentum space is capable
of an infinite number of discrete values which can be roughly determined,
while the distance R in coordinate space is not an observable quantity at all
This LACK OF SYMMETRY seems to me
very strange and rather improbable.”
Born’s principle of reciprocity
(„odwzajemnianie, wzajemność”)
Laws of nature are invariant under reciprocity transformation:
xk  pk, pk  -xk
H

x k pk ,
k
p

H
xk
xk , pl   i kl
Lkl  xk pl  xl pk
Arena of Physics
Macroscopic space on which (classical) processes are described
Usually adopted description:
arena = spacetime
Hamiltonian formalism:
independent position and momentum coordinates
CONJECTURE:
 use NONRELATIVISTIC PHASE SPACE as the arena
on which processes are to be described
 introduce constant [GeV/cm] to permit SYMMETRIC
treatment of momentum and position
Further arguments (nonrelativistic)
In classical Hamiltonian formalism – time occupies a distinguished
position – it is a parameter upon which p and x depend
In quantum mechanics p and x are operators,
while time is still a c-number parameter
Relativistic field theory unites relativity and quantum physics,
but does this in a formal way, argued by many to be unsatisfactory
(Wigner, Dirac, Chew, Finkelstein, Penrose)
Instantaneous reduction of a nonlocal state does not seem
to be in accord with the spirit of relativity
(instantaneous in which Lorentz frame?)
This highlights essential difference between time and space,
In spite ot these notions being united
in the concept of Minkowskian spacetime (description!)
Local Lorentz-equivalent frames are not fully equivalent physically
(background radiation defines a preferred frame)
Further physical arguments (GeV/cm)
1) Description in terms of spacetime followed from the observation
that Maxwell equations are form-invariant under Lorentz
transformations, which transform time  space,
with [c]=cm/s being the dimensional constant
permitting these transformations to be effected
2) String-like properties of hadrons
(with energy proportional to the length of the string)
suggests introduction of []=GeV/cm;
dual string model of 60’s-70’s: relevant constant is [’] = GeV - 2
(slope of Regge trajectories) (add [ħc] = GeV cm);
nowadays: string-like properties of hadrons - confined flux tubes,
here: more fundamental in origin
whether  is related to 1/’  1 GeV2 or m2Planck 10 38
GeV2
is irrelevant at the moment
Constants  (phase space, GeV/cm) and ħ (quantum, GeVcm)
suffice to set mass scale.
Thus, the problem of mass may be addresssed.
„Perfect figure” in phase space
Generalization:
2
p x
2
beyond Born reciprocity
beyond sphere in 3 dimensions
O(6)
Require
Poisson brackets (commutators)
to be form-invariant
U(1)  SU(3)
local?
„global”
Perhaps, but:
further conceptual shifts needed
Generators
SU(3)
F1  p1 p2  x1x2
F2  x1 p2  x2 p1
...
F5   x3 p1  x1 p3
...
F8  ...
U(1)
R=…
Reciprocity transformations,
Standard reflections
p  -x  - p
xp-x
Standard rotations + …
So far p and x treated completely symmetrically:
One cannot really say what is momentum and what is position
Standard mass:
momentum-position distinction
For individual objects separated by large distances:
energy of observed FREE objects (elem. particles) is defined by their
mass and momenta (either via relativistic or nonrelativistic formula)
THE STANDARD CONCEPT OF MASS
may be said to be directly ASSOCIATED
with the concept of momentum (p),
NOT position (x)
Recall: Born
The six-dimensional vector (p1,p2,p3,x1,x2,x3)
is divided into two triplets of canonically conjugated variables
in such a way that ONE of the triplets
is ASSOCIATED with the concept of „mass”
Generalization of the concept of mass
Is such a division into (p1,p2,p3) and (x1,x2,x3) unique?
Gen.momentum
SU(3)
gen.position
(p1,p2,p3)
(x1,x2,x3)
(p1,x2,-x3)
(-x1,p2,x3)
(x1,-x2,p3)
(x1,-p2,p3)
(p1,x2,-p3)
(-p1,p2,x3)
Odd # of p’s
SO(3)
SO(3)
Even # of p’s
Generalizarion through SU(3),
not U(1)
U(1)
Concept of mass
may be ASSOCIATED with
these 1 + 3 possibilities
Rotational (translational) invariance ?
Since each of three new choices violates rotational invariance –
- an object with mass associated with any one of these choices
CANNOT belong as individual object to our rotationally invariant
macroworld
However, these objects could belong to macroworld
as unseparable components of composite objects,
provided the latter are constructed in such a way,
that all appropriate invariance conditions are satisfied
CONJECTURE: QUARKS
Argument that strong interactions are rotationally etc. invariant is NOT a valid one:
What we ALWAYS see is the interaction of external probes (photons, W,Z bosons)
with colour-singlet currents. Colour-singlet current has quantum numbers of
a colourless meson, and from the point of photon, etc. behaves like a hadron –
not an individual quark (e.g. VMD)
In other words quarks have to conspire in such a way
that the resulting composite object – hadron - behaves in a „normal” way
Toy model
Apply SU(3) to Dirac equation
(simpler than non-rel. Schródinger)
(linearize p2+x2+m2)
H D  Ak pk  Bm
Ak   k   0   k  1   0
B    0   0 3  0
Bk  ..........   0   2   k
Ak , Al   Bk , Bl   2 kl
Ak , Bl   Ak , B  Bk , B  0
BB  1
H R  A1 p1  B2 x2  B3 x3  Bm
H G  B1x1  A2 p2  B3 x3  Bm
H B  A1 p1  B2 x2  B3 x3  Bm
H = HR+HG+HB
Rotational invariance restored
Charge conjugation
HD=Ak(pk- e A k)+Bm+e A 0
Explicit particles
Antiparticles through reinterpretation
c.c.: i - i, x x, t t, H - H, p - p, Ak A*k, B B*, A A, m m, e e,
-H’D= A*k(-pk- eA k)+B*m+e A 0
HD=Ak(pk+ eA k)+Bm-e A 0
HR=A1(p1- e A 1)+B2x2+B3x3+Bm
quark
C=-i 222
CB*C-1=-B
CA*kC-1=A k
CB*kC-1=B k
HR=A1(p1+e A 1)-B2x2-B3x3+Bm
antiquark
When particle goes into antiparticle,
physical quantities change as follows:
E E, ,p p, x - x, e  - e, m  m, (and i  - i)
Simple quark-antiquark system
H R  H R  A1 ( p1  p1 )  B2 ( x2  x2 )  B3 ( x3  x3 )  B  2m
Translation and rotation invariant after summation over quarks, antiquarks of all three types
Spin 0 and 1 (tensor product kl0)
Comments:
* additivity of quark charges
* additivity of quark masses
* appearance of a „string”
* intrinsic angular momentum of a quark: p1 x2 etc.
* objects exhibiting well-defined properties of one kind
do not have well-defined properties of another kind
* vague similarity to RISHON scheme of Harari (1979)
Two rishons (charge): T(+1/3), V(0)
e+ T T T
 VVV
u TTV TVT VTT
d VVT VTV TVV
Last transparency
Mirage only? I believe not…
CONCEPTS OF PHYSICS:
„Physics is based on well-founded concepts,
and its progress relies heavily on the creation of new
concepts.
Our understanding of Physics deepens as we gain new
perspectives on already existing notions.
Therefore, Physics continuously requires both new
concepts and new perceptions of the old ones.”