talking about nature

talking is in fact nothing but separating and grouping, they are the foundation of all concepts

this is especially evident in mathematics (a way of talking, in which the relations between concepts are clear,

and thus the foundation of any concept is determined)

the concept of a set (a group of parts) is the foundation of all mathematical concepts

the subtle point here is that, separating and grouping are circular concepts,

each defined using the other (yin and yang)

nature (everything) can't be a set

in other words, there is no set of everything, because at least it can't contain itself

even if we define sets, in a way that they could be members of themselves,

again a set of everything would not be possible (it's called Russell's paradox)

this is why:

suppose that we have a set named set1

another set named set2 contains those sets in set1 that are not members of themselves

it turns out that set2 cannot be in set1, here is the proof:

suppose that set2 is in set1, set2 is either a member of itself or it is not

if set2 is a member of itself,

then by its definition set2 must not be a member of set2, a contradiction

if set2 is not a member of itself,

since it is a member of set1 which is not a member of itself, set2 must be a member of set2,

again a contradiction

this shows that no set contains everything because at least it can't contain set2

it's the talking that must be contained in nature (eg in the form of a brain), not the other way around

nature, universe, reality, everything, god, being, or whatever we call it,

is not a group of parts, though it can be described as a group of parts approximately

ironically only the existence of (approximate) groups and parts in our brains,

allows us to talk about nature in general

so talking about nature, with approximate concepts, is possible, and indeed very useful,

it's called physics

quantum theory and general relativity (discussed in the following chapters) are

the best theories of physics we have at hand,

two different ways to talk about two different areas of nature

it's only at extreme areas namely Planck scales, that they meet each other,

and this meeting turns out to be disastrous, and at the same time, a useful guide to a unified theory

(as discussed in the chapter on Planck scales)

we will see that quantum theory and general relativity, when combined, imply that:

at Planck scales nature can't be described with parts and groups,

because at those scales the assumption that nature is a group of parts,

is not a good approximation any more

but even without mentioning quantum theory and general relativity,

we can predict the main aspects of the unified theory

since nature is not a concept, all concepts used in the unified theory must be unobservable

if something is observable, it means that it is a part of nature,

and it is observed through interaction with other parts

the following paragraph is quoted from "motion mountain" by Christoph Schiller http://www.motionmountain.net/

indeed there must be one and only one unobservable concept

if it was observable, the theory would not be final,

if there were more unobservable concepts, the theory would be fiction, not science …

in any unified theory, all the concepts that appear must be only approximately parts of the whole

thus we need an entity, describing nature which is not a set but which can be approximated by one

for example, the approximation should yield a set of space points and a set of particles,

but also, whenever we look at any part of nature, without any approximation,

we should not be able to distinguish it from the whole world

the simplest model would be a single entity which is extended and fluctuating,

reaches spatial infinity, allows approximate localization,

and thus allows approximate definition of parts and points

in more vivid imagery, nature could be described by some deformable folded and tangled entity,

a giant knotted amoeba

an amoeba slides between the fingers whenever one tries to grab a part of it

a perfect amoeba flows around any knife trying to cut it,

the only way to hold it would be to grab it in its entirety

however, for someone himself made of amoeba strands, this is impossible

he can only grab it approximately, by catching part of it,

and approximately blocking it, for example using a small hole,

so that the escape takes a long time

strand theory proposed by Christoph Schiller is an attempt for a unified theory of physics

(it's discussed in the last chapter)

still, no matter how we describe nature, we are here as pretty evolved parts of it

in fact we are so evolved that we can alter the path of evolution itself

the problem is that we are not still evolved enough to use this ability properly

we are going to create more suffering, and ultimately the destruction of all which has been achieved,

if we continue what we are doing now (excluding a few developed countries)

though it is possible that the evolution goes one step further,

leading to humans that can add to the beauty and richness of nature

see "a new earth" by Eckhart Tolle

spacetime is a real vector space with a Minkowski inner product

vector space and inner product are mathematical concepts whose precise definitions can easily be found online

a real vector space is a vector space which its underlying field is totally ordered, and has a complete metric

this means that the underlying field is isomorphic to real numbers

in a vector space with Minkowski inner product, all normal vectors (for which we have: v^2 = ±1),

are divided into distinct groups, each called a basis, for which we have:

b_i b_j = g_i_j

where:

g_ij = {

0 if i!=j

-1 if i=j=0

1 otherwise

}

vector components with respect to a basis are defined by:

v.i = v b_i

thus the inner product can be written as:

v1 v2 = g_i_j v1_i v2_j

by convention repeated indexes in a multiplication means summing over those indexes

and for convenience we use this notation:

v1_i v2_i' = g_i_j v1_i v2_j

transformations which leave the distance (ie |v2 - v1|) invariant,

are called symmetry transformations of spacetime

they can be written as ("d_i" is an arbitrary displacement):

v'_i = L_i_j v_j + d_i

where: L_i_k L_i_l = g_k_l

symmetry transformations of spacetime, form a group called spacetime symmetry group,

which is a ten'parameter non'Abelian Lie group

so the representations of spacetime symmetry group can be deduced,

using the representations of its corresponding Lie algebra,

plus the projective representations (since it's not a simply connected group)

this simple but usually overlooked question can actually be a guide to a unified theory,

by implying extended entities

3 is the only number of dimensions in which knots are possible

Zeeman proved that (considering topological isotopes, or even piecewise linear isotopes),

spheres can make knots only when the co'dimension is 2

(co'dimension is the difference between the dimension of the sphere and the container space)

there is another phenomena which i think is related to this: exotic smooth structures exist only on R^4

ie when "n" is not 4 then any smooth manifold homeomorphic to R^n is diffeomorphic to R^n

i think the origin of this phenomena is the failure of the Whitney trick,

which i think is because of co'dimension 2, between 2'disks used in Whitney trick, and the 4 dimensional space

point particles in a relativistic theory (in which instantaneous action at a distance isn't allowed),

lead to infinite self'field (thus external finite fields can't accelerate the particle)

"self'force on a classical point charge, Robert M. Wald"

http://www.math.utk.edu/~fernando/barrett/bwald1.pdf

in a relativistic theory only massless particles can be point'like

particles which can change velocity, ie massive charged particles, must have a non'zero size

note that this differs from the concept of a rigid body,

in fact in a relativistic theory, rigid bodies don't exist

the insides of particles must be considered special,

in the sense that even spacetime ceases to exist inside particles

considering gravity (discussed below), we can say this size is the Schwarzschild radius

furthermore, in general relativity, massive point particles (even with no charge) make no sense

fields must transform according to a representation of spacetime symmetry group

in other words fields must be a tensor or spinor on spacetime:

, scalar fields transform such that: f'(x') = f(x)

, vector fields: f'(x')_i = L_i_j f(x)_j

, higher order tensor fields: f'(x')_i_j_… = L_i_k L_j_l … f(x)_k_l_…

, spinor fields (note that the values here are actually complex numbers):

ψ'(x')_i = S_i_j ψ(x)_j

where: γ_i_j_k S_k_l = S_j_k L_i_m γ_m_k_l

where: γ_i_j_k γ_l_k_j + γ_i_j_k γ_i_k_j = 2 g_i_j

for higher order spinor gamma matrices are generalized into matrices (γ_i_j_…),

which are symmetric in any two tensor indexes

free fields are what we have when there is no interaction

a free field is a superposition of plane waves:

f(x)_i = a_i E^(I k_j x_j')

where "E" is the Euler's number and "I^2 = -1"

we define the mass of a field as: m^2 + k_i k_i' = 0

free field equation is an equation which every solution of it is a free field

free field equation for scalar fields:

∂∂f(x) + m^2 f(x) = 0

where: ∂∂ = ∂_i ∂_i'

(note that all formulas are written in Planck units in which: c=G=ħ=k=1)

free field equation for vector, tensor, or spinor fields with spin "s" (Joos-Weinberg equation):

I^2s γ_(i_1)_…_(i_2s)_j_k ∂_(i'_1) … ∂_(i'_2s) f(x)_k_… + m^2s f(x)_j_… = 0

for spin 1 we have:

∂∂f(x)_i + m^2 f(x)_i = 0

which in the case of a massless field, leads to Maxwell's equation for a free electromagnetic field:

∂∂f(x)_i = 0

for spin 1/2 we have the Dirac's field equation:

I γ_i_j_k ∂_i' ψ(x)_k - m ψ(x)_j = 0

deviations from free motion, for a field,

must always be accompanied by the deviation of another field from free motion

(kind of a generalized form of Newton's third law)

the exact mechanism of interaction (and thus the number of interacting fields) is

determined by the gauge symmetry of interaction

gauge (or local) symmetries are symmetries under spacetime dependent transformations

https://en.wikipedia.org/wiki/Gauge_theory

we can write a field equation that is invariant under

a local phase transformation (spacetime dependent phase transformation),

by introducing a vector field, to compensate for the variations due to the derivative term

I γ_i_j_k ∂_i' ψ(x)_k - m ψ(x)_j = E γ_i_j_k f(x)_i' ψ(x)_k

∂∂f(x)_i = E ~ψ(x)_k γ_i_k_j ψ(x)_j

where "~" is complex conjugate operator

spacetime is a geometric space, ie it has geodesics, which determine the motion of free objects in it

all free objects always move the same way, regardless of their mass and other properties

on the other hand, according to the equivalence principle of gravity,

objects near a massive (uncharged) body, move the same way, regardless of their mass and other properties

so falling objects determine the geodesics

this implies that in the presence of gravity, the geodesics are curved

gravity can't be described as a local (gauge) interaction with a field

to see why, imagine that you are in a closed box

there is nothing in nature which you can use to determine if you are experiencing gravity

in the case of EM field, you just need a charged particle

in other words, there is no charge for gravity

in non'relativistic physics, we have two choices to describe gravity:

, as an action at a distance

, as the curvature of space and time (NewtonCartan theory)

though it's ugly, because there is two separate metrics

https://www.physicsforums.com/insights/revival-newton-cartan-theory/

but in relativistic physics we can only describe gravity as the curvature of space'time

for a spacetime with a local Lorentz symmetry, there is a maximum curvature which can be reached,

when a body's mass fits inside the so called Schwarzschild radius (2 G m / c^2),

which is called a black hole

i think dark energy is the result of activities at the border of space,

so it's kind of a global feature of space, and can be interpreted as the cosmological constant

although cosmological constant is the same throughout space,

but during the growth of the universe, when its shape was different (eg during inflation),

the cosmological constant could have different values

there are discrepancies in a number of astrophysical observations:

, rotation velocities in galaxies doesn't decrease as distance from the center increases

, gravitational lensing studies, need a lot more mass than can be seen

, cosmic microwave background imprints, seems to show the existence of non'baryonic matter

, gravitational lensing studies of the galaxy cluster collisions (such as Bullet Cluster),

implies a component which does not follow baryonic matter

currently these observations are best described using dark matter,

a new kind of non'baryonic matter, which constitutes about 85% of all matter in the universe

but i think it's possible to explain these discrepancies as accumulative quantum effects of dark energy,

on the curvature of spacetime

quantum gravity + rotating atomic BoseEinstein condensate (superfluids) -> anti'gravity?

in the light of strand theory:

in macroscopic objects the direction of motion does not affect the extension strands

they are homogeneously spread aroud the object

but for quanta the extension strands lie in the direction of its motion

and depending on the rotation axis being in the direction of motion or the oposite of it,

the quanta is said to have up or down spin

what about superfluids?

observations like the double'slit experiment, show that:

universe is made of discrete but extended entities called quanta

quanta must be entirely and instantaneously created or destroyed in discrete units (like particles),

even though each unit is generally infinitely extended (like fields)

"Art Hobson, there are no particles, there are only fields"

https://arxiv.org/pdf/1204.4616.pdf

fields can be considered as the probability amplitude for interaction of quanta

this makes the measurement (which relies on interaction), a probabilistic procedure

the fact that everything, even measurement tools, have field behaviors, implies that:

measurement has an intrinsic uncertainty

uncertainty principle is a property of fields:

Δx Δk >= 1/2

Δt Δf >= 1/2

strong and weak interactions (unlike electromagnetism) have very short ranges,

thus in practice we always work with them in small scales,

containing a relatively small number of quanta

strong interaction is described by a SU(3) gauge symmetry

(3rd order special unitary group, the set of all 3 by 3 unitary matrices with unit determinant)

it implies 8 gauge quanta which even interact with each other,

leading to confinement, and thus the short range of strong interaction

weak interaction is described by a broken SU(2) gauge symmetry

the symmetry must be broken, otherwise weak interaction would have long ranges

to explain this symmetry breaking, we need another quanta called Higgs boson

the apparent violation of mirror inversion and time reversal symmetries in weak interactions,

is due to the asymmetry in the internal structure of particles (not an asymmetry of spacetime itself)

in other words processes that change the internal structure of particles,

seem to violate mirror inversion and time reversal symmetries,

because we are not considering the internal structure of particles

in other words, CPT symmetry is not violated

the important point here is that the decay of quantons actually shows us that,

quantons have some kind of internal structure

this section is quoted from "motion mountain" by Christoph Schiller

any particle that is smaller than its own Compton wavelength must be elementary

(the Compton wavelength of a particle is equal to the wavelength of a photon,

whose energy is the same as the mass of that particle)

if it were composite, there would be a lighter component inside it

this lighter particle would have a larger Compton wavelength than the composite particle

this is impossible, since the size of a composite particle must be larger than

the Compton wavelength of its components

however, an elementary particle can have constituents, provided that they are not compact,

as extended constituents have no localized mass

in the strand theory, elementary particles are (families of) tangles of strands

in other words, elementary particles are not the basic building blocks of matter, strands are

if particles could really be elementary, it would be impossible to understand their properties

in the strand model, particles are not really elementary,

but neither are they, in the usual sense, composed

particles are tangles of unobservable strands

in this way, the strand model retains the useful aspects of the idea of elementary particle,

but gets rid of its limitations

if one wants to think radically, the strand theory can be seen as,

eliminating the concepts of elementariness and of particle

a model for spin 1/2 is part of physics folklore since almost a century

any belt provides an example:

it is a famous exercise to show that such a model is indeed invariant under 4π rotation,

but not under 2π rotation

two such particles get entangled when exchanged, but get untangled when exchanged twice

particles can have spin 1/2, provided that they have tails going to the border of space

if the tails do not reach the border, the model does not work

spin 1/2 thus even seems to require extension

explaining black'hole entropy seems to demand extended entities too

at extremely small scales of spacetime, called Planck scales,

the mom'energy of quantum particles can be so high that black holes appear

therefore at Planck scales, nature is not observable, spacetime is indistinguishable from particles and fields,

they must be described by the same (unobservable) thing

the rest of this section is quoted from "motion mountain" by Christoph Schiller

general relativity and quantum theory contradict each other

in practice however, this happens only at Planck scales

whenever we combine general relativity and quantum theory,

the universe teaches us that it is not a set of parts

for this reason, any sentence or expression containing the term "universe",

is probably meaningless, whenever complete precision is required

combining quantum theory and general relativity leads to several important results on the description of nature:

, there is no conceivable way to prove that points exist,

as the smallest measurable distance in nature is the Planck length

, vacuum and particles mix at Planck scales,

as there is no conceivable way to distinguish whether a Planck sized region,

is part of a particle or of empty space

, matter, radiation and vacuum cannot be distinguished at Planck scales,

they are made of common constituents

, particles, vacuum and continuous space do not exist at Planck scales,

they disappear in a yet unclear Planck scale mixture

elements and sets must be abandoned

this radical conclusion is deduced from only two statements:

, the necessity of using quantum theory whenever the dimensions are of the order of the Compton wavelength

, and of using general relativity whenever the dimensions are of the order of the Schwarzschild radius

together, they mean that no precise description of nature can contain elements and sets

the difficulties in complying with this result,

explain why the unification of the two theories has not so far been successful

not only does unification require that we stop using space, time, and mass, for the description of nature,

it also requires that all distinctions, of any kind, should be only approximate

but all physicists have been educated on the basis of exactly the opposite creed

we need a description of nature that allows us to state that:

at Planck energy nothing can be distinguished from anything else

there is only one solution:

everything (or at least, what we call "everything") must be made of the same single constituent

there is an intriguing relation between Planck scales and cosmological scales,

they seem to pose the same challenges to their description

there is a tight relation between large and small scales in nature

there seems to be little difference (if any at all) between the universe and nothing

strands have no endings, thus there is actually only one ring tangled into itself

strands have no restriction other than themselves

a strand is everywhere at all times, unless when other strands restrict it

note that strand restriction is a collective process,

ie it's determined by the whole structure of a strand configuration

collective restrictions lead to a number of different rational (ie unknotted) tangles,

called elementary tangles, which are made of 3 or less strands

a photon is a twist on a strand, it moves (on average) at the speed of light

the twist turns around the strand, and the frequency of this rotation is the photon's frequency

massive elementary particles are tangles of two or three strands

in free space they move with a constant velocity (on average) less than the speed of light

charged particles have chiral tangles

due to collective restrictions, chiral tangles can absorb or emit photons,

and as a result change their (average) velocity

note that when a photon is absorbed, its original strand will be unrestricted,

and thus disappears into the border of space

this is also what happens when emitting photons

when a photon appears on a strand of a particle, and move away from it,

the strand between the particle and the photon will be unrestricted

some tangles are unstable and decay into stable ones

mirror tangles neutralize each other's restrictions, and allow the cores to be untangled,

which results in high frequency photons

virtual particles are temporary tangles, ie tangles which annihilate after a short time

mass is the amount of inter'tangledness, and thus it determines the inertial difficulty to move the tangle,

as well as the curvature of spacetime