talking about nature
1, talking about nature
talking is in fact nothing but separating and grouping; those are the foundation of all concepts
this is especially evident in mathematics, which essentially is 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 is not 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, there are two possibilities:
, 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,
it 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
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
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'll be introduced 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 in most parts of the world
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
2, spacetime, fields and particles
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, normal vectors (v^2 = ±1) can be grouped into distinct sets
each set is called a basis, and contains four vectors, for which we have (for i,j=0,1,2,3):
b[i] b[j] = g[i,j]
where:
g[i,j] = {
0 if i><j
-1 if i=j=0
1 otherwise
}
vector components (which are real numbers) with respect to a basis are defined by (for i=0,1,2,3):
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 indices in a multiplication means summing over those indices
an even more convenient notation would be:
v1 v2 = v1[i] v2[i']
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)
why 3 dimensions for space?
this simple but usually overlooked question can actually be a guide to the 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
fields
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'[i](x') = L[i,j] f[j](x)
, higher order tensor fields: f'[i,j,…](x') = L[i,k] L[j,l] … f[k,l,…](x)
, spinor fields (note that the values here are actually complex numbers):
ψ'[i](x') = S[i,j] ψ[j](x)
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 (γ[i,j,…]) into matrices which are symmetric in any two indices
free fields are what we have when there is no interaction
a free field is a superposition of plane waves:
f[i](x) = 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 whose every solution 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[k,…](x) + m^2s f[j,…](x) = 0
for spin 1 we have:
∂∂f[i](x) + m^2 f[i](x) = 0
which in the case of a massless field, leads to Maxwell's equation for a free electromagnetic field:
∂∂f[i](x) = 0
for spin 1/2 we have the Dirac's field equation:
I γ[i,j,k] ∂[i'] ψ[k](x) - m ψ[j](x) = 0
deviation of a field from free motion, must always be accompanied by the deviation of another field
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
we can write a field equation that is invariant under a local (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[k](x) - m ψ[j](x) = E γ[i,j,k] f[i'](x) ψ[k](x)
∂∂f[i](x) = E ~ψ[k](x) γ[i,k,j] ψ[j](x)
where "~" is complex conjugate operator
particles
point particles in a relativistic theory (where instantaneous action at a distance isn't allowed),
lead to infinite self'field (thus external finite fields can't accelerate the particle)
see "self'force on a classical point charge, Robert M. Wald"
thus in a relativistic theory, 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
3, gravity
spacetime is a geometric space, ie it has geodesics, which determine the motion of free objects in it
all free objects (fields and particles) 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 (Newton'Cartan theory)
though it's ugly, because there is two separate metrics
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 (2G m / c^2)
it's called a black hole
dark energy and dark matter
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
4, quantum theory
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)
see "Art Hobson, there are no particles, there are only fields"
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
strong and weak interactions (unlike electromagnetism) have very short ranges,
thus in practice we always work with them in small scales, with 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 quanta actually shows us that, quanta have some kind of internal structure
hints to extended entities
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
quantum gravity modification
is it possible to curve spacetime in small regions, without needing a huge mass?
quantum gravity, and rotating atomic BoseEinstein condensate (superfluids)
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?
5, Planck scales
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
6, strand theory
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
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