LOG#198. Bounds for Quantum Gravity?

Previous post was about 3…Now, we have a post about some “high 5”. These five?

Why 5 and not 3? I love Japan, ikigai fan? 😉

Well, numbers are cool. Now, 5 fingers on your hand…Oh, wait, you need a multipass for the 5th element, the perfect human being…

Look at this:

I am not a Canadian citizen, but I like some “Canadian ways”. Let me begin…


The most challenging unsolved issue for ANY theoretical physicist is likely the subject of quantum gravity (QG). No matter the success (partial and uncomplete!) of any current approach can deny we are yet lacking fundamental details. Superstring/M-theory, loop quantum gravity (LQG) and other minor areas or research like phenomenological quantum gravity or extended relativities need more data. Or, maybe, look at the actual knowledge from a different fashion.

In this post, I am going to remember my readers 4 (maybe 5?) bounds pointing out that something else, engaging and IMPORTANT, is lost in our current understanding of matter and energy. Also, information. Information as key concept is growing up, specially from Quantum Information Theory (QIP). The forthcoming quantum computing and robotics, likely the A.I. (Artificial Intelligence) could change the rules of the game forever, even for theorists and experimentalists!

1st bound. The Margolus-Levitin theorem.

Coming from the stunning world of Quantum Mechanics and its wacky-wimey rules…This powerful theorem states that the processing rate of ANY quantum computing event (or likely any other form of computation!) can NOT be higher than about 6\cdot 10^{33}operations/s/joule. Equivalently:

“Any quantum system of energy E needs at least a MINIMAL TIME (chronon) t_\perp.”


So, T\geq t_\perp (at least t_\perp!), and thus, \Gamma\leq 1/t_\perp, at most! That is, there is a maximal width

    \[\boxed{\Gamma\leq \dfrac{1}{t_\perp}=\dfrac{4E}{h}=\dfrac{2E}{\hbar \pi}}\]

Remark: Margolus-Levitin theorem is what I call the “orthogonal projectable chronon”, or the orthogonal chronon time acting on the projection postulate in any orthogonal quantum measurement.

2nd bound. Landauer’s principle.

Landauer’s principle states that there is a minimum possible amount of energy required to erase one bit of information, known as the Landauer limit:

    \[E_{min}=E_0=k_BT\ln 2 \]

For an environment or reservoir at temperature T, energy E=ST must be emitted into that environment if the amount of added entropy is S. For a computational operation in which 1 bit of logical information is lost, the amount of entropy generated is at least k_B\ln 2, and so, the energy that must eventually be emitted to the environment is

    \[\boxed{E\geq k_BT\ln 2}\]

3rd bound. The Bremermann’s limit.

In QUANTUM NOISE AND INFORMATION, H. J. BREMERMANN proposed (in 1965) a bound on channel capacity, that is, a limit on quantum speed! It is based on the requirements on relativity AND quantum mechanics, plus information theory. The capacity C of a band limited channel is given by a formula due to Shannon

    \[\boxed{C=\nu_m \log_2 \left(1+\dfrac{S}{N}\right)}\]

and where S and N are the signal and noise power.

Proposition (Bremermann’s limit, 1965). Channel capacity bound: “The capacity of any closed information transmission or processing system does not exceed mc^2/h bits per second, where m is the mass of the system, c the light velocity, h is the Planck’s constant.”

Note that it links the light barrier and the quantum of action barrier. Using the channel capacity from Shannon’s theory, for a signal that is at least EQUAL to the noise, you get that C\leq \nu_m \log_2 (2)=\nu_m=mc^2/h. It is dependent, of course, upon the validity of quantum mechanics, which, as physical theories in general, is subject to modification if empirical evidence contradicting the theory should be found. Sub-quantum theories could violate this bound. And quantum gravity?

Remark: The quantity h/c^2 is the mass equivalent of a quantum of an oscillation of one cycle per second. c^2/h\approx 1.36\cdot 10^{50} bits/s/kg. The proposition above can be restated as follows. Information transmission is limited to frequencies such that the mass equivalent of a quantum of the employed frequency does not exceed the mass of the entire transmission or computing system. Put in a different way: each bit transmitted in one second requires a mass of at least the mass equivalent of a quantum of oscillation of one cycle per second. Interestingly, it seems that the forthcoming new definition of kilogram is going to use h/c^2 as the new pattern of mass! Bremermann’s also discusses the Landauer’s limit (without such as name) in the paper cited above.

4th bound. Caianiello’s maximal acceleration limit.

Trying to derive quantum mechanics from a phase-spacetime geometry, Caianiello’s main discovery is linked to the so-called maximal acceleration principle, that it states that for any mass (energy) there is a maximal acceleration (gravitational field, if you keep the equivalence principle valid in some way):


and thus

    \[\boxed{a\leq A_M=2\dfrac{Mc^3}{\hbar}}\]

Do you see the connection with 2 (maybe the three?) previous bounds?

5th bound. The Bekenstein’s bound.

From the darkest and deepest mysterious black holes, following interesting thermodynamical arguments, J. Bekenstein derived the following bound.  For any gravitating system with energy E and size R, the next entropy (information) limit holds:

    \[\boxed{S\leq \dfrac{2\pi k_B ER}{\hbar c}}\]

in joules per kelvin degrees. Equivalently, in bits, you get

    \[\boxed{I\leq \dfrac{2\pi ER}{\hbar c\ln 2}}\]

Using E=Mc^2 and plugging R in meters, M in kilograms, you get:

    \[\boxed{I\leq \dfrac{2\pi MR}{\hbar \ln 2}\approx 2.58\cdot 10^{43} MR}\]

Example: For a human brain, with a mass about 1.5kg and volume 1260cm^3, assuming spherical brain form (spherical cow jokes are allowed!), it yields

    \[I\leq 2.6\cdot 10^{42}bits\]

as maximal information stored in the brain, and it equals to the maximal information necessary in order to mimic an average human brain down to the quantum mechanical level. Assuming the brain is quantum, it should have about 2^I quantum states, or equivalently N\leq 10^{7.8\cdot 10^{41}} states! Note that the Bekenstein bound is SATURATED by black holes (with entropy S_{BH}=k_BA/4L_P^2).

The question is: are these 5 bounds independent or are they only 5 aspects of a same deep principle of the missing quantum gravity? Think by yourself about it! Are they 5 or a single one in disguise?

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