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Transcript of Fuzzy Topology, Quantization and Long-distance Gauge Fields SSss S.Mayburov Hep-th 1205.3019 Quantum...
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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)
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SSss of List of Quantization Formalisms:
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
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S Sets, Topology and Geometry
Example: Fundamental set of 1-dimensional Euclidian geometry is
(continious) ordered set of elements X = { xl } ; xl - points
ji xx , . .i j j ix x or x x
X xi xj
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ST - total set; ST = X P u
P u = { uj } - partial ordered subset
Xx xi
u1
D Dx
Fu1 ~ xk , Dxxk
ji xx , it can be also: xi ~ xj - incomparabilty relation
pPPppPartial ordered set – P u
B Beside
u1 ,xk are incomparable (equivalent) ST elements
X = { x l } - ordered subset;
XDxi Example: let’s consider interval -
u1 - incomparable point
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Partial ordering in classical mechanics
Beside geometric relations between events there is also causal relations
t
x
events - causal set: { e j }
Their time ordering corresponds to :
t1 ≤ t3 ≤ t2
e3
e1
e2
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Partial ordering in classical mechanics
Beside geometric relations between events there is also causal relations
t
x
events - causal set: { e j }
Their time ordering corresponds to :
t1 ≤ t3 ≤ t2
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
e3
e1
e2
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Partial ordering in special relativity
The causal relations acquire invariant, principal meaning due to light cone
t
x
event set: { e j }
Their time ordering is : e1 ≤ e3 ≤ e2
Let’s describe also causal relations between events, then:
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
e3
e1
e2
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If f j - fuzzy points, then: if Ffi ~ fk
Ffi , fk , wi,k >0 , F ={f i} - set of fuzzy points; Σ wjk =1
Fuzzy ordered set (Foset) - F ={ fi } (Zeeman, 1964)
. . . . . .
_ xi
f1
X
EExample: ST - total set
ST = X F; F={ f1 }
X – ordered subset
w1i
Ffi ~ xk , wi,k >0 , Σ wjk =1
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Continuous fuzzy sets
EExample: ST - total set ; ST = R1 F;
F={ f1 } – discrete subset of fuzzy points
X – continuous ordered subset; X = { xa }
w
fi , x ; w(x) > 0;
x
1)( dxxw
ax
is fuzzy parameter
ST is fuzzy space
x~
w(x) dispersion characterizes f coordinate uncertainty
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x
Classical mechanics:
In 1 dimension the particle m is ordered point x(t)
its state: | m } = { x(t), vx(t), … }
describes m trajectory
Massive particle is fuzzy point m(t), its coordinate is fuzzy parameter
its state: |g} = {w(x,t), …. ? }
x(t0)
x(t)
Fuzzy mechanics:
w(x,t0 ) w(x,t )
)(~ tx
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X
Fuzzy evolution in 1-dimension
Fuzzy state | g } : | g}= {w(x), q1 , q2 … }
Beside w(x,t), another |g} parameter is - average m velocity
we shall look for | g} free parameters assuming that they can be
represented as real functions qi(x,t)
)(tv
v
w(x,t0 )
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hhh
Fuzzy Evolution Equations
Let’s consider free w(x,t) evolution :
),( txftw
Suppose that w(x,t) evolution is local and so can be described as:
where f - arbitrary function
then from w(x) norm conservation:
dxxf )( 0),(),(
dxtxwt
dxtxtw
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hhh
Fuzzy Evolution Equations
Let’s consider free w(x,t) evolution :
),( txftw
where f - arbitrary function
then from w(x,t) norm conservation:
dxxf )( 0),(),(
dxtxwt
dxtxtw
if to substitute : )(xfxJ
then: 0),(),( tJtJ
),( txxJ
tw
If this condition is fulfilled, then :
- 1-dimensional flow continuity equation
Suppose that w(x,t) evolution is local and so can be described as:
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x1 X
w (x,t)
v(x1)
in 1-dimension m flow can be expressed as: )()()( xwxvxJ
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
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; k is theory constant
c is arbitrary real number
cdvkxx
)()(
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:
),()()( wiexwxg
for m state g = { w(x), v(x)}, we substitute v(x) by related parameter:
where is arbitrary real function
for any the set of g(x) constitutes complex Hilbert space H
),( w
),( w
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We suppose that observables are linear, self-adjoint operators on H
Correct g(x) ansatz should give all values to be independent of c value To derive it it ‘s enough to consider flow (current) operator
Then :
It follows that the only g(x) ansatz which give value to be
independent of c value is written as :
This is standard quantum-mechanical ansatz
)()()( xiexwxg
Q
Q
)(ˆ xJ
)(2
)(*
*
x
gg
x
gg
k
ixJ
)(xJ
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We start with linear operator for fuzzy dynamics,
later the nonlinear operators will be considered also
For free m motion H should be invariant 0],ˆ[
dx
dH
It means that H can be only polinom:
j can be only even numbers, j = 2,4,6,… because noneven j contradict to X -reflection invariance
n
ii
i
i xcH
2
ˆ
gHt
gi ˆ
Evolution Equation for Fuzzy state g(x)
relative to space shifts so:
H
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Equation for operator
iet
wtw
itg
i )(
)(wvxt
w
ffr
From flow equation:
2
2
211
xw
kxw
xktw
it follows:
hence: iewxg )(
xkxv
1
)(and
H
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iet
wtw
itg
i )(
)(wvxt
w
ffr
From flow equation:
it follows:
hence: iewxg )(
xkxv
1
)(and
and:
gHt
gi ˆ
n
ii
i
i xg
c2
n
i
i
i
i
i ewx
c2
)(
iex
wki
xw
xki
tw
tg
i )2
(2
2
in
n
ni e
xwicewH )...ˆ
2
2
211
xw
kxw
xktw
From comparison of highest α derivatives it follows n=2;
fuzzy state g(x,t) obeys to free Schroedinger equation with mass m = k !
kc
21
2
H Equation for operator
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Nonlinear evolution operator
Bargman - Wigner theorem (1964) : 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 ) : g(t0 ) → g’(t)
i.e. if all pure states evolve only to pure ones,
and is defined unambiguously, then is linear operator
Yet fuzzy pure states g(x,t) evolve only to pure ones, so their evolution should
be linear
tU
tU
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
tU1ˆ tU
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IIWe didn’t suppose that our theory possesses Galilean invariance,
in fact, Galilean transformations can be derived from fuzzy dynamics,
if to suppose that any RF corresponds to physical object with m
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Dirac equation :
Its fuzzy space: F = { } is double (entangled) fuzzy structure
)( mp
ti
zsr ~~3
Starting from it and demanding that w(x,t)>0
It results in the linear equation for 4-spinor which is
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: ),( txxJ
tw
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Fuzzy space: F = { } generates Hilbert space H of
complex functions g(x,t) which describe particle’s evolution in F
Observables are linear, self-adjoint operators on H
particle’s momentum:
commutation relations follow from
geometrical and topological properties, so in this approach
fuzzy topology induces such operator algebra
What about and ?
Modified fuzzy space F supposedly can be underlying structure for them
x~
dxd
ipx
Q
ixpx ]ˆ,ˆ[
]ˆ,ˆ[ yx ]ˆ,ˆ[ yx pp
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Fuzzy States and Secondary Quantization
e
Example: Fock free quantum field { nk } in 1-dimension
- particle numbers can be regarded as fuzzy values
each particle possesses fuzzy coordinate
- double fuzzy structurexnk
~~
kn~
x~
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Free Schroedinger equation:
is equivalent to system of two equations for real and imaginary parts of
Interactions in Fuzzy Space
gHt
gi ˆ
2
2
211
xw
mxw
xmtw
- flow continuity equation I )
2
2
2
)(211
xmxw
wmt
II )
t
g
here I) is kinematical equation, it describes w(x,t) balance only
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Interactions in Fuzzy Space
gHt
gi ˆ
2
2
211
xw
mxw
xmtw
I )
2
2
2
)(211
xmxw
wmt
II )
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 is quantum phase
+ Hint
Free Schroedinger equation:
is equivalent to system of two equations for real and imaginary parts of t
g
- flow continuity equation
),( tx
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Particle’s Interactions in Fuzzy Space - Toy model
2
2
211
xw
mxw
xmtw
I )
2
2
2
)(211
xmxw
wmt
II )
Let’s consider interaction of two particles m1 , m2 , their states: g1,2 (x,t)
+ Hint
g1 g2
In geometrical framework it assumed that m1 react just on m2 presence
i. e. if , then It means that there is
term in Hint = Q1 Q2 F(r 12 )= , where Q1 ,Q2 are particle charges.
02,1 p 0int H
x
int)( Ht free
)( 121 rAQ o
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ttt t
H (t)
H (t+Δt )
H (t+2Δt )
α
Hence this is shift of α(x,t) subspace of Et = H - fiber bundle
So that each H (ti ) is flat but their connection C(ti , tk ) depends on Hint
T
Hence: tHtrAQ int012010 )(
tH
)()( 1201 rAQtt free
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(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 can interact nonlocally;
Fermions with fuzzy coordinates can nonlocally interact with bosonic field.
12~x
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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
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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)
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Fuzzy Correlations
We exploit fuzzy state:
Yet more correct expression is :
)()()( xiexwxg
)}',(),({ xxkxwg
here: )'()()',( xxxxk
Correlation of g(x) state between its components x, x’ appears due to particle fuzziness on X axe, it defines g(x) free evolution:
Such ansatz corresponds to quantum density matrixMore complicated correlations can appear in fuzzy mechanics.
xxxk
xv
}',()(
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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,
even if ; yet for
and
02,1 p
0int H
)()( 12011221int rAQrFQQH
this is deformation of m1 fiber bundle H
2
2
2
)(211
xmxw
wmt
+ Hint
T
Hence: )()( 1201 rAQtt free
1
21
0 2mp
H
02 p
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Particle’s Interactions in Fuzzy Space
So this toy-model is equivalent to Electrodynamics if Aμ is massless field
suppose that m1 is described by Klein-Gordon equation with E>0
and m2 is classical particle, i.e. m2
grAQgmrt
gi )(01
212
2
If in initial RF: then:
For Lorenz transform RF to RF’ with velocity
')'('')]'(''
['
0121
21 grAQgmrAQ
ri
tg
i
v
only such vector field makes transformation consistent
where : 201
'v
vAA
A
'gg
and 2001
1'
vAA
02 p
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D
Fuzzy state for fermion (iso)doublet
Consider fermion (iso)spin doublet of mass m0
Fuzzy parameters: ; particle’s state :
Fuzzy representation: g =
2SU
- (iso)spin vector
- linear representation
z
s
zszyx ~,~,~,~ ),( zsrg
)(),(),(),( rrrrw
from Jordan theorem ψs evolution should be linear
2
1)(
rg s
ss H
ti s
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D
Nonabelian case: Yang-Mills Fields
Consider interaction of two femions m1 , m2 of doublet and suppose
that their interaction is universal, so that at it gives
int 1 2 1,2( )H ks s f r
- isospin vectors
To agree with global isotopic symmetry, two isospin vectors in nonrelativistic limit can interact with each other only via this ansatz:
Hence:
- Yang-Mills
s1 s2
For nonrelativistic QCD limit it gives:
But it supposes that local gauge invariance follows from global isotopic symmetry :
2( )
gf r
r
2SU
)()( 122120 rfsrA
2,1s
)( 2,101int rAH
02,1 p 0int H
2SU
2
1)(
rs
and so :
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Yang-Mills Interactions
So we obtain standard Yang-Mills Hamiltonian for particle-field interaction
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 Yang-Mills field
grAQgmrt
gi j
j
)(01212
2
in initial RF:
For Lorenz transform RF to RF’ with velocity
')'('')]'(''
['
0121
21 grAQgmrAQ
ri
tg
i jjj
j
v
only such vector field makes transformation consistent
where : 201
'v
vAA j
j
jA
'gg
and 2001
1'
vAA
)(0 rA
02,1 p
2SU
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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 fiber bundleT
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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
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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
1c 13712 e
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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
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Particle’s Interactions in Fuzzy Space - Toy model
2
2
211
xw
mxw
xmtw
I )
2
2
2
)(211
xmxw
wmt
II )
Let’s consider interaction of two particles m1 , m2 , their states: g1,2 (x,t)
+ Hint
g1 g2
In geometrical framework it’s natural to assume that Hint is universal,
i. e. if , then It means that
there is term in Hint ~ Q1 Q2 F(r 12 ), where Q1 ,Q2 are particle’s charges.
02,1 p 0int H
x
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Relativistic Evolution Equation
n
ii
i
i xccH
22
2
20
we suppose that fuzzy state: iexg )(
and: || w || =1
),()( 1 fxw ),()( 2 fxJ
gHt
gi ˆ
0)( x
If we look for 1st order evolution equation:
where:
If twIII then for
the unique solution is: n=2 , i.e. Schroedinger evolution
n
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Δ1
X
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
Example : Boltzman diffusion
w(x,t0 )w(x,t )
'),'(),'(),( 00 dxtxwttxxGtxw
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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 tb
Constraint: each event time t1 in one RFis mapped to the point on time axe in other RF t’1
e1 ~ e3
jdivtw
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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
dxd
k1
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m particle’s state |g} = { w(x), α(x) }
The problem : to find dynamical |g} representation g(x,t) :
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:
)()()( xiexwxg
gHt
gi ˆ
H
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Generalization of event time mapping
one can substitute points by δ-functions
ta ~ e3
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FuzzFuzzy dynamics – space symmetry restoration
as the possible fundamental low of dynamics
X
w0
w(t)
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
E Global symmetries are important in quantum physics
Free particle m:
w0(x): space shift symmetry
is broken
w0
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m motion can be described as the space symmetry violation for |g}
w(x)
α(x)
X
m velocity: <Vx> = < >dxd
X
d α(x) is analog of quantum phase
dxddxd
fffffff
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Δ1 Δ2
X
v(x1 ) v(x2 )
henchence w flow equation can be written as:
dxdv
wdxdw
vxj
tw
x
dtw
xwxv )(
)(1
)(
jdivtw
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hello
ggggggkggghggghfh
FuzzFuzzy mechanics – space symmetry restoration
as the general low of free motion