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[Phys-l] IS PLANCK”S CONSTANT SCALE DEPENDENT? Typo corrections.






IS PLANCK”S CONSTANT SCALE DEPENDENT?
In Quantum Field Theory, space time is treated as a passive background
structure, the stage where particle interactions take place. However, we know this
picture is incorrect. General Relativity teaches us that space time is not a
passive background but an active player in physical processes. In fact this
interaction of space time which we call the force of gravity is Universal.
Nothing in the Universe is exempt from having a relationship with gravity.
The only reason we are able to have a successful theories which ignore these
facts is because Gravity is a very weak force. The incredible weakness of
gravity gives us one of our unsolved puzzles in physics, which goes under the
name of the Hierarchy problem.
However, there are phenomena as well as theoretical issues where gravity can’
t be ignored. This occurs when the accumulation of matter is so great that
gravity become a powerful force or equivalently we deal with physics at a small
enough scale where the effects of gravity become comparable to the other
forces of nature.
Currently our best theoretical tool to incorporate gravity into our quantum
theories is the semi classical formalism. This formalism is based on the
premise that we can leave gravity classical while at the same time we bring in
the quantum fields of all the other interactions.
In the semi classical model we can modify the Einstein equation to;
G_mu,nu= kappa* < T_mu,nu>_psi
Where G_mu,nu is the classical Einstein tensor and < T_mu,nu>_psi is the
expectation value for the stress energy operator given the state of the
collective quantum fields. The success of this approach suggests that some form of
induced gravity model can serve as an effective field theory for gravity at
large scale.
Nevertheless, we have good reason to expect that gravity, like the other
interaction fields in nature is a
described by a quantum theory. That is, the curvature fluctuations in the
Einstein tensor come in quanta called gravitons.
Developing such a theory has proved immensely difficult. While there are
several candidate theories which show various degrees of promise no totally
convincing theory exists.
In spite of the limited success of these Quantum theories of gravity, they
all share the premise that space time is quantified just like matter; that is
space time too comes in atoms. This premise is strongly supported when we
look at the thermodynamics of black holes, where gravity becomes very powerful.
Given a sufficient quantity of matter gravity creates a causal boundary
between the collected mass and the rest of the Universe. This sets a strict limit
on the information content of any volume of space time, which we call the
called the Bekenstein bound named after the physicist who first proposed this
limit. Of course having finite information content for any volume of space
suggests that space time comes in some form of atomic units.
If space time does come in atomic units, then we must generalize the
Heisenberg uncertainty principle to incorporate the effects of the gravity field.
This gravity effect shows up in T duality in string theory and the quantization
of geometry in Loop quantum Gravity and the other related canonical quantum
gravity models.
The standard model uncertainty equation-inequality is
Delta (P)*Delta(x) = > hbar/2
Where P is momentum and x is the position.
However, when we incorporate the effects of gravity (which in the Super
string theory involves the duality between wound and unwound string vibration
modes, a very elegant formalism) we must generalize the uncertainty relationship
to;
Delta (P)*Delta(x) => (hbar/2)* (2+2*L_u^2*Delta (P) ^2/hbar^2)
Where L_u is the distance scale where the effects of gravity equals the
other interactions, either the string scale or the Planck scale for the canonical
models.
We can define
Alpha_p= L_u^2/hbar^2
As the momentum scaling constant which in SI units equals 2.336E-2
So we can write
Delta (P)*Delta(x) => (hbar)* (1+*alpha_p*Delta (P) ^2)
We see that the uncertainty relation ship is now scale dependent. So we can
introduce a scale dependent Planck term which we can call h_g
We see that
h_g= hbar*(1+alpha_p*Delta (p) ^2)
We can also include this scale dependence in the energy –time uncertainty
equation-inequality where the new scaling constant is
Alpha_E= _L_u^2/ (hbar^2*c^2) = 2.595E-19 in SI units
So that we can write
Delta (E)*Delta (t) => h_g= hbar*(1+ Alpha_E*Delta (E) ^2)
We can see that
(1+ Alpha_E*Delta (E) ^2) = 2 at the Planck scale.
So that hbar scales by a factor of two down to the Planck Scale.
Given this new relationship we can calculate the space time atomic unit
directly.
Since we have
Delta(x)*Delta (P) => hbar*(1+ L_u^2*Delta (P) ^2/hbar^2)
We can take L_u to be the Planck scale so that we have
P= {x+sqrt[x^2 -4*_L_plk^2] /2*L_plk^2}*hbar
Based on this we have a fundamental atomic unit of space where we have a
space time wick rotation when;.
X_atom= 2*L_plk
Given the Bekenstein Bound for the information content of a given volume of
space, which we know, is directly proportional to the surface area of the
given volume we have;
S=A_surface/X_atom^2 = A_surface/ (4*L^2_plk)
This is the correct Bekenstein equation for the information content of a
black Hole.
Therefore the actual effective Planck constant is scale dependent and we can
define the correct atomic units of space time using the generalized
uncertainty principle which is required when the gravity field is to be considered.
This introduces a modified dispersion relationship so that looking at the
wavelength of a particle;
Lambda= h_g/P = hbar*(1 +Alpha_p*P^2)/P = hbar/P_g
And we get the new Energy eigenvalue
E= (N+1/2)*h_g*w = (N+1/2)*hbar*w*(1+ Alpha_E*^2)
Of course at most energy scales the scaling constants can be set to zero
restoring the standard quantum relationships.
Bob Zannelli









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