Condensed Matter Model

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Overview SM fermions SM gauge fields Gravity Quantization Open problems Other theories

Titlepage

A condensed matter model giving SM fermions,gauge fields, and a metric theory of gravity

Ilja Schmelzer

email: [email protected], www: http://ilja-schmelzer.deFoundations of Physics, vol. 39, 1, p. 73 (2009), arXiv:0908.0591
http://find/http://goback/


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Modern physics

The situation in fundamental physics

Established theories:

General Relativity (GR):Metric g(x), 0 , 3, x = (x0, . . . , x3);Standard Model (SM) of particle physics:

Twenty four Dirac fermions;Twelf gauge fields(eight gluons, three weak bosons W+, W, Z, one phonon);Higgs field? not yet observed!

Problems:

Observation: dark matter, dark energy, inflation;

Quantization of gravity;

Unification of SM and gravity;

Explanation of SM;


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The new ether

The new ether: A lattice of cells in R3

Classical (Newtonian) framework: absolute space R3 and time R;

Space is filled with a medium (the ether), which consists of:

A lattice of elementary deformable cells;

Some unspecified material between the cells;


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The aim

The aim: A theory of everything

All particles are quantum effects of waves (similar to phonons).

Fermions: Basic oscillations exactly all SM fermions.Strong gauge fields: Material between cells U(3)c.

Weak gauge fields: Lattice deformations U(2)L U(1)R.Smax = S(U(3)c U(2)L U(1)R) = GSM U(1).Gravity: Density , velocity vi, stress tensor ij g;

All particles observed until now are identified. The model explains: why three generations, why the gaugegroup acts identical on them, why three colors, why electroweakdoublets, why all parts of a doublet have the same color, why weakcharge does not depend on color and baryon number, whyright-handed neutrinos are inert.

O i SM f i SM fi ld G i Q i i O bl O h h i


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Standard Model Fermions

Standard Model Fermions

Twenty four Dirac fermions:3 (3 + 1) (2 4)

Three identical generations of eight Dirac fermions; In each generation four doublets:

Quark doublets: (up-quark, down-quark), . . .Each quark in three colors three doublets;One lepton doublet: (electron, neutrino), . . .

Each fermion described by four complex fields;

Left-handed and right-handed components;

Particles and antiparticles;

Dirac equation: it = iii + m

Square-root of wave equation: 2t = (i 2i + m2)

O i SM f i SM fi ld G it Q ti ti O bl Oth th i


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The cell lattice

The cell lattice: Fermions

State of cell: affine deformation yi

= i

jxj

+ i

0from standard reference cell at origin O.

Aff(3) = {i R, 1 i 3, 0 3}.

Configuration space of the whole lattice:Aff(3)(Z3): {i(n1, n2, n3) : Z3 Aff(3)}

Phase space: Aff(3)

C(Z3):

{i(n1, n2, n3)

C

}SM: Each complex lattice function = one electroweak doubleti: generation index, : color index, > 0: quarks, = 0: leptons


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Doubling

The doubling effect

Low energy solutions of2t = (2h + m2)smooth only on mod 2 sublattices!

we need eight continuous functionsto describe one oscillating lattice function

= (1, 2, 3), i {0, 1}

i

(2n1 + 1, 2n2 + 2, 2n3 + 3) = i

(n1h, n2h, n3h)

Electroweak doublets: C(Z3) = {(n1, n2, n3)} C8(R3)Fermions: Aff(3) C(Z3) = {i(n1, n2, n3)} Aff(3) C8(R3)



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Advantage of non-relativistic approach

Comparison with relativistic approach

dim = 3: Spatial lattice 8 staggered fields 123 (ni, t).dim = 4: Spacetime lattice 16 staggered fields 0123 (n).

Kogut-Susskind staggered fermions.

dim = 3: it(x, t) = iii(x, t) + m(x, t); (x, t) C8dim = 4: i(x) = m(x); (x) C16

dim = 3: geometric Dirac operator on C

(R3) with metric ij

dim = 4: Dirac-Kahler equation on C (R4) with metric dim = 3: Two Dirac fermions = Electroweak doublets of SM.dim = 4: Four Dirac fermions

No interpretation in SM.



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Geometric interpretation

Geometric interpretation

i

i is geometric Dirac operator on differential formsC

(R

3

)Staggered geometric discretization on cell complex:

Interpolation rule: Value of k-form is integral over k-cell.

Staggered lattice: different for even and odd nodes.

Doubling: 1 field per lattice node eight fields ofC

(R

3

).



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g g y Q p p

Geometric rotations and spin

geometric rotations and spin

The generators of geometric rotations ij appear as combination ofspinor rotations ij and isospin rotations Ik:

ij = ij 2iijkIk

If the mirror rotates 180o,

and the light sources dont,

the image rotates 360o.

What we observe (the image),

behaves like a spinor, but really we have only a fixed background.

The fixed isospin direction I3 is analogon of fixed light sources.



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g g y Q p p

Standard Model Gauge Fields

Standard Model Gauge Fields

Maximal group on 24 Dirac fermions: U(48) (dim = 482

).Realized: A twelf-dimensional subgroup SU(3)c SU(2)L U(1)Y:Strong interaction: SU(3)c eight gluon fields ;

act only on quarks; changes only color; same inside doublets;

Weak interaction: SU(2)L three bosons W+, W, Z;

acts only inside doublets; preserved doublets;acts only on left-handed parts;independent of color and baryon/lepton charge; massive;

Electromagnetic interaction: U(1)em one photon ;

preserves doublets; same for left- and right-handed parts;different for quarks and leptons;different for upper and lower particle in a doublet;

All gauge fields:

preserve generations and act identically on them;

do not act on right-handed neutrinos;


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Euclidean symmetry

Euclidean symmetry

Left E(3) action on Aff(3)(Z3): The lattice is rotated as a whole.

Postulate: GSM commutes with left E(3) action.

rotations: i(n) ijj(n) Rotations rotate the three generations of the SM.

GSM acts inside the generations (no mass terms!). GSM acts on all three generations in same way.

translations: i0(n) i0(n) + ti shift in some direction in the leptonic sector of the SM.

GSM leaves right-handed neutrino invariant.



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Symplectic structure

GSM preserves symplectic structure

complex structure =Euclidean structure gkld

kdl + symplectic structure kldk dl

Phase space = natural symplectic structure: dpk dqk.Postulate: Symplectic structure kl invariant

Problem: No natural Euclidean structure in phase space!We start with an arbitrary Euclidean structure ., .0. We consider compact gauge groups Haar measure. Averaging gives invariant Euclidean structure:

, =G

d(g) g, g0This gives the invariant complex structure: iak = gkllma

m

GSM should be unitary group.



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Wilson gauge fields

Wilson gauge fields

Up to now, the gauge group may be as large as U(15) (16 Weylfermions in each generation, only 1 preserved).

Which of them can be realized on the lattice?

Imagine irregularities between the cells.This has some influence on the equation.We compensate it by some correction term,which acts on the phase space of one node.

= Wilson gauge field.Group G acts pointwise. But electroweak doublets are (n) C.Thus, G has to have the same charge on the whole doublet.

=

G

U(3)c =

SU(3)c

U(1)B



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Lattice deformations as gauge fields

Lattice deformations as gauge fields

Properties of correction terms for lattice deformations:

They change the equation for (n) C,

preserves electroweak doublets,

no color and baryon charge dependence. Gweakmax = U(3) SU(2)L

Further improvement is possible: We have tolook how the lattice Dirac equation changes.

nhin nhi(n +

j aj(n)n+hj)



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Shift operators on the lattice

Shift operators on the lattice

nhin nhi(n +

j aj(n)n+hj)

2in = n+2hi becomes 2i123 (x) = 123 (x + dxi)

in = n+hi mixes different 123 (x).

i = fi(1, 2, 3)i with 2i = 2i, [i, HD] = 0: i 25Ii

Generators: {1, i 25Ii, ij 2ijkIk, 123 5}

This generates a maximal gauge group U(2)L U(2)R.Translational symmetry U(2)L U(1)R or U(2)R U(1)L.

Charge IR =1+5

2 (I3 12 ) on all doublets.



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Neutrality of the vacuum state

Neutrality of the vacuum state

Volovik: . . . equilibrium homogeneous ground state of condensedmatter has zero charge density . . . electroneutrality is the necessary

property of bulk metals and superconductors; otherwise thevacuum energy of the system diverges faster than its volume.

Ground state (Dirac sea) should be neutral

Sum of charges (trace) of all fermions should be zero.

Postulate: Gauge group should be special group

Gmax = S(U(3)c U(2)L U(1)R) GSMHypercharge: U(1)Y

S(U(1)B

U(1)L

U(1)R)



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The upper axial gauge field

The upper axial gauge field

Only one additional upper axial gauge field IU = 5(I3 12 ).Variant I

U= IU +

15

2 I3 =1+5

2 I3 5

2 commutes with GSM.

Gmax = S(U(3)c U(2)L U(1)R) = GSM U(1)UGmax = S(U(3)c U(2)L U(1)R) anomalous!

BRST quantization forbids anomalous gauge fields.

GSM is maximal non-anomalous subgroup of Gmax.

But: No lattice gauge symmetry BRST cannot be used anyway.If gauge symmetry is not required, anomaly may be unproblematic.



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Gravity

General relativity

Metric theories of gravity: g(x), 0 , 3, x = (x0, . . . , x3)

Clock time along path :

g((s))(s)(s)ds

Lagrangian: L = Lgravity(g, . . .) + Lmatter(g, matter)

Einstein equivalence principle (EEP): Lmatter covariant;

Strong equivalence principle (SEP): Lgravity covariant;

Lgravity(g) = f(R)g

Preserving lowest order terms: LGR = (R

2)

g



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Ether interpretation for metric theories of gravity

Ether interpretation for gravity

For metric theories of gravity in harmonic coordinates X(x),the ADM decomposition for the foliation T(x) = X0(x)gives a natural condensed matter interpretation:

g00 g = g0i

g = vigij

g = vivj ij

harmonic condition:X

=(

g

g

) = 0continuity equation: t + i(v

i) = 0

Euler equations: t(vi) + i(v

ivj ij) = 0



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A variant of the Noether theorem

A variant of the Noether theorem

Theory depends on preferred frame defined by coordinates X

(x).The preferred coordinates X(x) are four scalar fields on R4. There should be Euler-Lagrange equations for the X(x):

S

X

=L

X

L

X,

L

X,

. . .implicit coordinate dependence: v0(x)

= explicit coordinate dependence: v(x)X0,(x)

Noether theorem:Translational symmetry X(x) X(x) + c gives

S

X=

L

X, . . .

= 0.



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Derivation of the general Lagrangian

Derivation of the Lagrangian

Postulate: The Noether conservation laws are proportional tocontinuity and Euler equations:

S

T= (t + i(v

i))

S

Xi= (t(v

i) + i(vivj ij))


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Derivation of the general Lagrangian

Derivation of the Lagrangian

Postulate: The Noether conservation laws are proportional tocontinuity and Euler equations:

S

T= (t + i(v

i))

S

Xi= (t(v

i) + i(vivj ij))

SX

= (gg) = X

for = ii, = 00. General solution =

particular solution + general sol. of homogeneous problem

L = 12 X,X

,g

g + LGR(g) + Lcov(g, matter)



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Physical predictions of the theory

Physical predictions:

Exact Einstein Equivalence Principle.

The Einstein equations of GR in some natural limit.

No black hole singularity gravitational collapse stops beforehorizon formation, leading to a stable gravastar.

No big bang singularity big bounce before big bang.

Flat universe preferred as the only homogeneous universe.

> 0 X0(x) timelike = no closed causal loops.

Note: 0 possible, but parts with < 0 unphysical.


O i


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Overview

Quantization

We know how to quantize condensed matter theories. Canonical Quantization

What has to be quantized is the fundamental theory - the

lattice of cells and the material between them. No separate quantization of gauge fields and gravity!

Analogon: Quantum condensed matter theory not result ofphonon field quantization.

Volovik: No cosmological constant problem.Problem: How to obtain anticommuting fermion operators forcanonical quantization of real fields?


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Obtaining a spin field from a real field


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Obtaining a spin field from a real field

How to get a Z2-valued field from a real field

Lagrangian for a relativistic scalar field withZ

2-degenerated V():L = 12 ((t)2 (i)2) + 2

2 2 4! 4

Regularization: Lattice Z3 R3.i(x)

1

h

(n+1

n)

t(x) 12 (tn+1 + tn)Canonical quantization!

1H

1

0H

0 e

3

2

2

H2

0

H0Low energy domain generated by 0/1(n) in each node:

Pauli matrices: (in)2 = 1,

im,

jn

= 2imnijk

kn .

H = c0n

3n + c1n,i

1n1n+hi

+ c2n,i

2n2n+hi


The isomorphism between spin field operators and fermion operators


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The isomorphism between spin field operators and fermion operators

Spin fields are not fermion fields!

Spin field operators on different nodes commute.

Fermion opeators on different nodes anticommute:

{m, n} = mn, {m, n} = {m, n} = 0.

But isomorphism exist!

1n = n + n,

2n = i(n n), 3n = i1n2n.

1/2n =

1/2n

m>n

3m, 3n =

3n,

1/2n =

1/2n

m>n

3m, 3n =

3n.

This is known in Clifford algebra theory: ClN,N(R)

= M2N(R).


Choice of the order


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Choice of the order

Choice of the order

Isomorphism nonlocal, not natural, depends on order.

dim = 1: Natural order i-local Hamiltonian local in i too.dim > 1: Hamiltonian nonlocal in i: in

in =

in

in

nm>n3m

We apply a -local projection: (in

in) =

in

in

3n.

Error H H depends on order;Our choice of order:

Justification: It exactly preserves

inin+hk

= 1 12 ((1 k)in)2:(

in

in+hk

) = inin+hk

= 1 12 ((1 k)in)2


Transformation of the Hamilton operator


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H = c0n

3n + c1n,i

1n1n+hi

+ c2n,i

2n2n+hi

Special case: c0 =m2 , c1 = c2 = 14h .

H = 12hn,i

nn+hi(nn+hi nn+hi) + m2n

nn nn

H is our lattice Dirac operator!

itn = [H, n] =1

2hi

n

n+hi(

n+hi

nhi) mn

2tn =1

4h2i

(n+2hi 2n + n2hi) m2n = (2h + m2)n.


What is missed?


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What has been left to future research?

Equations of motion for gauge fields;

Quantization of gauge fields;

Symmetry breaking:

Masses of fermions;Masses of weak gauge fields;Is there a Higgs sector?

What about the additional gauge field?

Renormalization (renormalization group equations);Closer connection between SM and gravity;

Quantization of gravity;


quantum effects of gauge fields


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q g g

Quantum effects of gauge fields

U(2)L U(1)R has no exact gauge symmetry on the lattice.Standard (BRST) quantization requires exact gauge symmetry!

We need no separate quantization procedure, but a derivationof quantum properties of the effective gauge-like fields.

We have a definite Hilbert space structure.

Manifest Lorentz invariance of BRST has been given up.

No factorization of gauge degrees of freedom they are

physical!We need an evolution equation for them:Lorenz gauge A = 0.

We need gauge-breaking energy terms.


Symmetry breaking


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Spontaneous symmetry breaking

Mass matrix violates rotational symmetry,

fermion quantization violates translational symmetry.

=

spontaneous symmetry breaking necessary.

A main argument for Higgs is invalid:

Broken theory has no spatial isotropy.

Maybe there is no Higgs?

Maybe the bosonic partners of fermions play their role?


Gauge field masses


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Gauge field masses

Observation: Weak fields massive (short range);

Continuous theory: Gauge invariance gauge field massless;SM: Fundamental theory gauge-invariant; Broken symmetry;

Our theory: Nice correspondence:

Nine gauge fields (U(3)c) have exact lattice gauge symmetry. Nine gauge fields (SU(3)c U(1)em) massless. Weak gauge fields have no lattice gauge symmetry. Weak gauge fields are massive.

Unknown mechanism U(3)c SU(3)c U(1)em.


Fermion masses


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Fermion masses

Quantum condensed matter: Phonons

acoustic phonons: k 0 E 0 (massless);Exactly three, associated with translations;

optical phonons: k

0

E

E0 > 0 (massive);

All other;

Our model: More close to translations lower masses;Neutrinos: almost massless;Exactly three, associated with translations;

Leptons: mass larger than neutrinos, smaller than quarks;

Quarks: larger masses than leptons;

Upper quarks have higher masses than lower quarks;


Foundations of quantum theory


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Foundations of quantum theory

Violation of Bells inequalities:Realism + Einstein causality = Bells inequalities;Realism + violation of BI = Necessity of preferred frame.

For a realist, the violation of Bells inequalities is an indirectobservation of the preferred frame.

Support for realistic interpretations of quantum theory:

Fermion quantization follows canonical quantization.

= Canonical constructions of hidden variable theories(de Broglie Bohm, Nelson) applicable to fermions.

A theory of relativistic gravity with preferred frame.= Removes an important argument

against interpretations with a preferred frame.


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Problems of field-theoretic approach to GR quantization


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Problems of field-theoretic approach

g g(x) =

( + h(x))

Examples: String theory, GR in harmonic gauge,Logunovs RTG with massive graviton.

Causality violations: Tachyonic solutions, where the lightcone of g(x) greater than that of .

Absolute causality not violated in this case.

The harmonic condition (g

g) = 0 is a first order

constraint which is problematic for quantization.

In Lagrange variables xi(xi0, t) the continuity equationvanishes, and the Euler equation becomes a second orderequation.


Comparison with string theory


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Comparison with string theory

One basic lattice model; 10300 possible universes;Basic model predicts SM particle context almost exactly; No physical prediction at all;Extremal simplicity;

Extremal complexity;Hidden preferred frame: independent evidence (even indirectobservation) given by the violations of Bells inequality;

Hidden dimensions with Minkowski space structure:

no independent evidence; 10 man-years of private research; > 20, 000 man-years;One paper accepted by Foundations of physics;

Tens of thousands of published papers;


End


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Thank you very muchfor your attention

email: [email protected], www: http://ilja-schmelzer.deFoundations of Physics, vol. 39, 1, p. 73 (2009), arXiv:0908.0591

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Condensed Matter Model