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2, spacetime, particles and fields
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2, spacetime, particles and fields
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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)
"https://en.wikipedia.org/wiki/Representation_theory_of_the_Lorentz_group"
"https://en.wikipedia.org/wiki/Spinor"
= why 3 dimensions for space?
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
= particles
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"
"https://en.wikipedia.org/wiki/Regularization_(physics)#Classical_physics_example"
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
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
= gravity
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
= dark matter
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?
"https://en.wikipedia.org/wiki/Ning_Li_(physicist)"
"https://en.wikipedia.org/wiki/Eugene_Podkletnov"
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?