Download Fuzzy topology, Quantization and Gauge Fields

Fuzzy Topology, Quantization and Long-distance Gauge Fields
SSss
S.Mayburov
Hep-th 1205.3019
Quantum space-time and novel geometries:
Discrete (lattice) space-time – Fundamental Plank length lp
( Snyder, 1947; Markov,1948)
Noncommutative geometry (Connes, 1991)
Supergeometry (Rothstein,1986)
Hypothesis: Some geometries can induce also novel formalism
of quantization (S.M.,2008)
of List of Quantization Formalisms:
SSss
i) Schroedinger quantization: S state Ψ is vector (ray) in Hilbert space H
ii) Operator algebra (C* algebra) : S observables constitute nonassociative
operator algebra U
( Segal, 1948)
iii) Symbols of operators
(Berezin, 1970)
… etc.,
…) Geometric Formalism
-
why not ?
Possible applications : Quantum Gravity
S
Sets, Topology and Geometry
Example: Fundamental set of 1-dimensional Euclidian geometry is
(continious) ordered set of elements

x,
i x
j
X = { xl } ;
xl - points
x
xo
.x
x
i
j. r
j
i
xi
xj
X
pPPppPartial ordered set – P u
B Beside x
i x
j , it can be also: xi ~ xj - incomparabilty relation
u1
D Dx
u1
xxi
X = { x l } - ordered subset;
- incomparable point
X
ST - total set; ST = X  P u
P u = { uj } - partial ordered subset

X
Example: let’s consider
interval - Dx
i
x
Dx
Fu1 ~ xk , 
k
u1 ,xk are incomparable (equivalent) ST elements
Partial ordering in classical mechanics
Beside geometric relations between events there is also causal relations
t
e2
events - causal set:
{ e j}
e3
Their time ordering corresponds to :
e1
t1 ≤ t3 ≤ t2
x
Partial ordering in classical mechanics
Beside geometric relations between events there is also causal relations
t
e2
events - causal set:
{ e j}
e3
Their time ordering corresponds to :
e1
t1 ≤ t3 ≤ t2
x
if one event dynamically induces another
:
e1 → e2
If to describe also causal relations between events:
e1 ≤ e2 ;
e1 ~ e3 ;
e2 ~ e3
So e3 is incomparable in causal terms to e1 and e2
Partial ordering in special relativity
The causal relations acquire invariant, principal meaning due to light cone
t
e2
event set: { e j }
e3
Their time ordering is :
e1 ≤ e3 ≤ e2
e1
Let’s describe also causal relations between events, then:
x
e1 ≤ e2 ;
e1 ~ e3 ;
e2 ~ e3
if e3 is outside the light cone, then other causal
relations between e3
and e1,2 can’t exist
Fuzzy ordered set (Foset) - F ={ fi }
(Zeeman, 1964)
If f j - fuzzy points, then: if Ffi ~ fk
Ffi , fk ,
 wi,k
>0 , F ={f i} - set of fuzzy points; Σ wjk =1
EExample: ST - total set
ST = X  F;
w1i
F={ f1 }
X – ordered subset
f1
.
_
.
.
xi
.
.
.
X
Ffi ~ xk , wi,k >0 ,
Σ wjk =1
Continuous fuzzy sets
EExample: ST - total set ;
ST = R1  F;
F={ f1 } – discrete subset of fuzzy points
X – continuous ordered subset; X = { xa }
 fi , x ;  w(x) > 0;
   xa  
 w( x)dx  1
w
~
x is fuzzy parameter
ST is fuzzy space
x
w(x) dispersion characterizes f coordinate uncertainty
Classical mechanics:
In 1 dimension the particle m is ordered point x(t)
its state:
| m } = { x(t), vx(t), … }
describes
m
x(t)
trajectory
x(t0)
Fuzzy mechanics:
w(x,t0 )
w(x,t )
x
Massive particle is fuzzy point m(t), its coordinate ~
x (t) is fuzzy parameter
its state: |g} = {w(x,t), …. ? }
Fuzzy evolution in 1-dimension
Fuzzy state | g } :
| g}= {w(x), q1 , q2 … }

v 
w(x,t0 )
X
Beside w(x,t), another
we shall look for

|g} parameter is v(t) -
| g} free parameters assuming
represented as real functions
qi(x,t)
average m velocity
that they can be
Fuzzy Evolution Equations
Let’s consider free w(x,t) evolution :
Suppose that w(x,t) evolution is local and so can be described as:

w
where f - arbitrary function


f(x
,t)

t
then from w(x) norm conservation:




w 
(
x
,
t
)
dx

w
(
x
,
t
)
dx

0
f (x)dx 



t

t




hhh
Fuzzy Evolution Equations
Let’s consider free w(x,t) evolution :
Suppose that w(x,t) evolution is local and so can be described as:

w
where f - arbitrary function


f(x
,t)

t
then from w(x,t) norm conservation:




w 
(
x
,
t
)
dx

w
(
x
,
t
)
dx

0
f (x)dx 



t

t




if to substitute :

J
f(x) then:

x
J
(

,
t
)

J
(

,
t
)

0
If this condition is fulfilled, then :

w 
J

 (
x
,t) - 1-dimensional flow continuity equation

t

x
hhh
v(x1)
w (x,t)
x1
in 1-dimension m flow can be expressed as:
X
J
(
x
)v
(
x
)
w
(
x
)
where v(x) is w flow velocity, it is supposed to be independent of w(x)
m state g(x) = { w(x), v(x)} – it’s observational g state representation
for m state g = { w(x), v(x)}, we substitute v(x) by related parameter:
;

(
x
)

k
v
(

)
d

c

x


k is theory constant
c is arbitrary real number
so m state is: g = { w(x), α(x)}, then g ~ g(x) = g1 (x) + ig2(x)
we suppose that g(x) is normalized, so that g(x)g* (x) = w(x),
yet g(x) is still arbitrary complex function:
i

(

,
w
)
g
(
x
)

w
(
x
)

e
where  ( , w) is arbitrary real function
for any
 ( , w)
the set of
g(x)
constitutes complex Hilbert space H
We suppose that observables
Correct
Q̂
are linear, self-adjoint operators on H
g(x) ansatz should give all
Q
values to be independent of
c value
To derive it it ‘s enough to consider flow (current) operator Jˆ ( x )
Then :
i
g *
* g
J ( x) 
(g
g
)
2k
x
x
J (x )
It follows that the only g(x) ansatz which give
independent of c value
is written as :
g ( x)  w( x)  ei ( x)
This is standard quantum-mechanical ansatz
value to be
Evolution Equation for Fuzzy state
i
g(x)
g
ˆg
H
t
We start with linear operator
Ĥ
for fuzzy dynamics,
later the nonlinear operators will be considered also
For free m motion H should be invariant
relative to space shifts so: [ Hˆ ,
d
]0
dx
i

ˆ 
ci i
It means that H can be only polinom: H
x
i2
n
j can be only even numbers, j = 2,4,6,…
because
noneven j contradict to X -reflection invariance
Equation for operator
From flow equation:
Ĥ

w 

 (
wv
) and

t

x

1

v
(x
)
k
x

2
w
1


w
1
it follows: 


w
2

t
k

x

x
2
k
x

ffr i

g
w i

g
(
x
)

w
ehence: i

(
i

w
)
e

t

t

t
Equation for operator
From flow equation:
Ĥ

w 

 (
wv
) and

t

x

1

v
(x
)
k
x

2
w
1


w
1
it follows: 


w
2

t
k

x

x
2
k
x

ffr i

g
w i

g
(
x
)

w
ehence: i

(
i

w
)
e

t
and:
g ˆ
i H
g
t

t

t
i
ig n 
i

ci

c
(w
e
)

i
i
i
x i
x
i2
2 
n





2
n

g
i


w
i 
i

i

i

i
(
w
 
w
)
e
ˆw
2 
H
e

...

ic
w
)
e
n
n

t

t
k

x

x
2
k

x

x
1
From comparison of highest α derivatives it follows n=2; c2 
2k
fuzzy state g(x,t) obeys to free Schroedinger equation with mass m = k !
Nonlinear evolution operator
Bargman - Wigner theorem (1964) Û
: t
g(t0 ) → g’(t)
If arbitrary pure state evolves only into the pure state and module of scalar
product for all pure states |< ψ|φ >| is conserved, then the evolution operator
is linear.
Jordan (2006): If for all g(t0 )
: Û t g(t0 ) → g’(t)
i.e. if all pure states evolve only to pure ones,
and
Uˆ t1 is defined unambiguously, then Û t
is linear operator
Yet fuzzy pure states g(x,t) evolve only to pure ones, so their evolution should
be linear
g(x,t) are normalized complex functions,
we obtained free particle’s quantum evolution on Hilbert space H
Origin of the results : w(x,t) flow continuity is incompatible with the
presence of high x - derivatives in
g evolution equation
We didn’t suppose that our theory possesses Galilean invariance,
II
in fact, Galilean transformations can be derived from fuzzy dynamics,
if to suppose that any RF corresponds to physical object with
m


Assuming the formalism extension to relativistic invarince,
Let’s assume that the formalism can be extended to relativistic invariance
In this case the flow continuity equation is fulfilled:

w 
J

 (
x
,t)

t

x
Starting from it and demanding that w(x,t)>0
It results in the linear equation for 4-spinor which is




i

(
p

m
)
Dirac equation :

t
Its fuzzy space: F = {
~
~
r3
sz }
is double (entangled) fuzzy structure
Fuzzy space: F = {
~
x}
generates Hilbert space
H
of
complex functions g(x,t) which describe particle’s evolution in F
Observables Q̂ are linear, self-adjoint operators on H
particle’s momentum:
ˆ x , xˆ]  i
[p
px i
d
dx
commutation relations follow from
geometrical and topological properties, so in this approach
fuzzy topology induces such operator algebra
What about
[ xˆ, yˆ ] and [ pˆ x , pˆ y ] ?
Modified fuzzy space
them
F
supposedly can be underlying structure for
Fuzzy States and Secondary Quantization
Example: Fock free quantum field { nk } in 1-dimension
e
n~k
- particle numbers can be regarded as fuzzy values
each particle possesses fuzzy coordinate
~ ~
n
x - double fuzzy structure
k
~
x
Interactions in Fuzzy Space
Free Schroedinger equation:
g
i
 Hˆ g
t
is equivalent to system of two equations for real and imaginary parts of


g
t
2

w
1


w
1
- flow continuity equation
I)


w
2

t m

x

x
2
m

x
II )


2
 1

w
1

2


(
)
2

tm
w

x2
m

x
here I) is kinematical equation, it describes w(x,t) balance only
Interactions in Fuzzy Space
Free Schroedinger equation: i
g ˆ
 Hg
t
is equivalent to system of two equations for real and imaginary parts of


g
t
2

w
1


w
1
- flow continuity equation
I)


w
2

t m

x

x
2
m

x
II )


2
 1

w
1

2


(
)
2

tm
w

x2
m

x+ Hint
here I) is kinematical equation, it describes w(x,t) balance only
Hence particle’s interactions can be included only in II)
But it supposes their gauge invariance, because  ( x, t ) is quantum phase
Particle’s Interactions in Fuzzy Space - Toy model


2

w
1


w
1

I)


w
2

t m

x

x
2
m

x
II )


2
 1

w
1

 
2


(
)

(
) free  H int
+ Hint
2

tm
w

x
2
m

x
t
Let’s consider interaction of two particles m1 , m2 ,
g1
their states: g1,2 (x,t)
g2
x
In geometrical framework it assumed that m1 react just on m2 presence
i. e. if

, then
p
0
1
,2
term in Hint =
H
0It means that
int
there is
Q1 Q2 F(r 12 )= Q1 Ao (r12 ) , where Q1 ,Q2 are particle charges.



 


( )

Q
A
(
r
)
free
1
012

t 
t
Hence: 



Q
A
(
r
)

t



H

t
0
1
0
12
0
int
ttt
t
Ht
H (t+2Δt )
H (t+Δt )
H (t)
α
Hence this is shift of α(x,t) subspace of
So that each H
Et = H T-
(ti ) is flat but their connection
fiber bundle
C(ti , tk ) depends on Hint
(Non)locality of interactions on fuzzy space
We supposed that in our model the interactions are local,
yet such theory is generically nonlocal,
two particles with fuzzy coordinates ~
x12 can interact nonlocally;
Fermions with fuzzy coordinates can nonlocally interact with bosonic field.
Conclusions
1.1. Fuzzy topology is simple and natural formalism for introduction
of quantization into any physical theory
2. Shroedinger equation is obtained from simple assumptions,
Galilean invariance follows from it.
3. Local gauge invariance of fields corresponds to dynamics
on fuzzy manifold
Fuzzy Topology, Quantization and Gauge Fields
S. Mayburov
hep-th 1205.3019
To explore quantum space-time it can be important to describe
Quantum physics and space-time within the same mathematical formalism
Proposed approaches:
i)
To extend quantum operator algebras on space-time description:
Noncommutative Geometry (Connes, 1988), etc.;
Applications to Quantum Gravity (Madore, 2005)
ii) To extend some novel geometry on quantum physics:
Possible approach: Geometry equipped with fuzzy topology (S.M., 2007)
Fuzzy Correlations
We exploit fuzzy state:
i

(
x
)
g
(
x
)

w
(
x
)

e
Yet more correct expression is :
here:

g


{
w
(
x
),
k
(
x
,
x
'
)}

k
(
x
,
x
'
)

(
x
)

(
x
'
)
Correlation of g(x) state between its components x, x’ appears due to
particle fuzziness on X axe, it defines g(x) free evolution:

k
(x
,x
'}
v
(x
)

x
Such ansatz corresponds to quantum density matrix
More complicated correlations can appear in fuzzy mechanics.
Particle’s Interactions on fuzzy manifold
let’s suppose that two identical particles m1 , m2 repulse each other
Deflections of m1 motion induced by m2 presence at some distance,
2

p
0
even if p
; H
0yet for H0  1
1
,2
int
2m
1
and

p
0
2
H

Q
Q
F
(
r
)

Q
A
(
r
)
int
1
2
12
1
0
12


2
 1

w
1

2


(
)
2

tm
w

x
2
m

x+ Hint

 


(
)

Q
A
(
r
)
Hence:
free
1
012

t 
t
this is deformation of m1 fiber bundle
H
T
Particle’s Interactions in Fuzzy Space
suppose that m1 is described by Klein-Gordon equation with E>0
and m2 is classical particle, i.e. m2 
If in initial RF:

p
0
2
then:
2

g

2
i 

m
g

Q
A
(
r
g

1
1
0)
2

t 
r
For Lorenz transform RF to RF’ with velocity

v

g
'



22
i
[
i

Q
A
'
(
r
'
)]

m
g
'

Q
A
'
(
r
'
)
g
'

1
1 1
0

t 
r
'


v
1
A
'

A
A
'

A
where :
and


0
0
0
2
2
1

v
1

v
only such vector field

A
makes
g
g' transformation consistent
So this toy-model is equivalent to Electrodynamics if
Aμ is massless field
Fuzzy state for fermion (iso)doublet
Consider
D fermion SU 2 (iso)spin doublet of mass m0

~
~
~
~
Fuzzy parameters: x
;
particle’s state : g
,y
,z,s
(r,sz)
z
 z
Fuzzy representation: g = w
(
r
),
(
r
),
(
r
),
(
r
)




 
1
g


r
)
  - linear representation
s(



2


i s H

s

t
from Jordan theorem
ψs


s

 - (iso)spin vector
s
evolution should be linear
Nonabelian case: Yang-Mills Fields
Consider interaction of two femions m1 , m2 of SU 2 doublet and suppose
D

0
p
0
that their interaction is universal, so that at 1
it gives H
,2
int
1
 
s(r) 
2


s1, 2
- isospin vectors
To agree with global isotopic symmetry, two isospin vectors in nonrelativistic limit
can interact with each other only via this ansatz:
s1
s2
H
k
s
sf(r
)
in
t
12
1
,2


Hence: A
(
r
)

s
f
(
r
)
0
12
2
12



A
(
r
)
and so :H
- Yang-Mills
int
1
0
1
,
2


For nonrelativistic QCD limit it gives:
But it supposes that local gauge invariance follows from
global isotopic SU 2 symmetry :
g
f (r)  2
r
Yang-Mills Interactions
Suppose that m1 is described by Klein-Gordon equation with E>0
and m2 is classical particle, i.e. m2
; and
so m2 generates classical SU 2 Yang-Mills field
in initial RF:

p
0
1
,2

A0(r)
2


g 
2
i  2

m
g

Q
A
(
r
)
g
1
1
0
j

t 
r
j
For Lorenz transform RF to RF’ with velocity

v

g
'
 2 2 
i
[
i 
Q
A
'
(
r
'
)]

m
g
'

Q
A
'
(
r
'
)
g
'
1
j
j
1
1
0
j

t 
r
'
j
  1
  v
j
A
'0
A
A
where : A
and

0

j'
0
2
2
1

v
1

v
only such vector field

Aj
makes
g
g' transformation consistent
So we obtain standard Yang-Mills Hamiltonian for particle-field interaction
We derived free particle’s dynamics from some geometrical
and set-theoretical axioms.
Is it possible to describe particle’s interactions in the same
framework ?
We exploited m states: g = { w(x), α(x)} defined in
flat Hilbert space
H
Analogously to general relativity, in geometry euipped with
fuzzy topology the interactions induce
the deformation of
H
and
H T
fiber bundle
Fuzzy Mechanics and Gauge Invariance
In QFT local gauge invariance is postulated.
We shall argue that in Fuzzy mechanics it can be derived
from global isotopic invariance
Such formalism need only three axioms, whereas standard QM
is based on seven axioms
i)
Particles are points of fuzzy manifold, their states are characterized
by two parameters |g } = { w(x,t), v(x,t) }
ii)
Observables Q are supposed to be linear, self-adjoint operators on H
iii) Reduction (Projection) postulate for result of Q measurement
In this model Plank constant is just coefficient, which connects
x and px scales, so Relativistic unit system gives its natural description
c
1
1
e2 
137
Content
1. Geometric properties induced by structure of fundamental set
2. Partial ordering of points, fuzzy ordering, fuzzy sets
3. Fuzzy manifolds and physical states
4. Derivation of Shroedinger dynamics for fuzzy states
5. Interactions on fuzzy manifold and gauge fields
For simplicity only 1+1 geometry will be considered
Particle’s Interactions in Fuzzy Space - Toy model
I)
II )


2

w
1


w
1


w
2

t m

x

x
2
m

x

H
2
 1

w
1

2


(
)
+
2

tm
w

x2
m

x
Let’s consider interaction of two particles m1 , m2 ,
int
their states: g1,2 (x,t)
g2
g1
x
In geometrical framework it’s natural to assume that Hint is universal,
i. e. if

, then
p
0
1
,2
there is term in Hint ~ Q1
H
0 It means that
int
Q2 F(r 12 ), where Q1 ,Q2 are particle’s charges.
Relativistic Evolution Equation
we suppose that fuzzy state:
i
 
(x
)
0
g
(
x
)

e


w
(
x
)

f
(,)
1


J
(
x
)

f
(,)
2
and: || w || =1
If we look for 1st order evolution equation:
i
g
ˆg
H
t
2
i


c
c
where: H

0
2
i
2
i
x
i
2 
n
If twIII
then for
n
the unique solution is:
n=2 ,
i.e. Schroedinger evolution
Fuzzy evolution and linearity
We don’t assume that w(x,t ) is linear function of w(x’,t0 )
which is standard assumption of classical kinetics
w
(
x
,
t
)

G
(
x

x
'
,
t

t
)
w
(
x
'
,
t
)
dx
'
0
0

w(x,t0 )
w(x,t )
Δ1
X
Example : Boltzman diffusion
Clocks, proper times and partial ordering
In special relativity each dynamical system has it is its own proper time,
there is no universal time, so proper time connected with clock dynamics
of given reference frame (RF)
ta


w

div
j

t
tb
Constraint: each event time t1 in one RF
is mapped to the point on time axe in other RF t’1
e1 ~ e 3
It can be transformed to symmetric complex g representation in H
It can be proved that only such g(x) representation
corresponds to consistent dynamics
g(x) → { w(x), v(x)} is unambiguous map
α(x)
is analogue of quantum phase
1
k
d
dx
m particle’s state |g} = { w(x), α(x) }
The problem : to find dynamical |g} representation g(x,t) :
g ˆ
i  Hg
t
Ĥ is dynamical operator (Hamiltonian)
We have two degrees of freedom: g = { w(x), α(x) },
it can be transferred to:
g(x) = g1 (x)+ ig2 (x)
such that g1 (x), g2 (x) = 0, if w(x) = 0
This is symmetric g representation
example:
i

(
x
)
g
(
x
)

w
(
x
)

e
Generalization of event time mapping
one can substitute points by
δ-functions
ta
~ e3
FuzzFuzzy dynamics – space symmetry restoration
as the possible fundamental low of dynamics
E
Global symmetries are important in quantum physics
Free particle m:
w0
w0(x): space shift symmetry
is broken
w(t)
X
Low of fuzzy evolution :
space symmetry of m state |g} is restored as fast as possible :
So free m evolution U(t) : w0(x)
const at very large t
Interactions: to restore space symmetry two identical particles e1 , e2
should repulse each other
m motion can be described as the space symmetry violation for |g}
fffffff
α(x)
X
w(x)
X
m velocity: <Vx> = <
d
d
dx
α(x) is analog of quantum phase
>
henchence w flow equation can be written as:

w

j
dw
dv




v
w

t 
x dx
dx
v(x1 )
v(x2 )
Δ1
X
Δ2


w

div
j

t

1x
w
v
(
x
)

  ()
d
w
(
x
)

t


FuzzFuzzy mechanics – space symmetry restoration
as the general low of free motion
hello
ggggggkggghggghfh