Derivation of Bremermann's Limit

Question

The argument for Bremermann's limit, as I understand it, goes something like this:

Begin with the time-energy uncertainty principle, as

$\Delta E \Delta t \ge h$ (Other papers use other factors of $h$/$\hbar$, but Bremermann (1962) uses $h$, so let's stick with that.)

Substitute $\Delta E \le mc^2$ to get: $mc^2 \Delta t \ge h$, and rewrite as:

$1 / \Delta t \ge mc^2 / h$

Which sets a maximum limit on the clock rate of any physical process. You usually see this phrased as "The maximum computing rate is $1.36 \cdot 10^{50}$ bits per second per kilogram", and oftentimes this is phrased as the time it would take to e.g., strongly solve chess.

Is there a reason $\Delta E \le mc^2$? For instance, could we imagine $\psi = c_1 \psi_1 + c_2 \psi_2$ with $E_1 \approx 0$, $|c_2|^2 \ll 1$, and $E_2 \gg mc^2$? It seems this would satisfy $\langle H \rangle \le mc^2$ but leave $\langle H^2 \rangle - \langle H \rangle ^2$ unbounded.

Is such a state physically impossible (moreso than a computer that uses its entire mass-energy for computation constantly)? For a free particle, I can see that measuring the energy as $E_2 = p^2/2m > mc^2$ yields $p > mc$, which maybe suggests a velocity greater than $c$, but:

1. Can you get around that with a potential $V \approx mc^2$?
2. This is using non-relativistic QM for something in the relativistic domain. In non-quantum relativity, $p > mc$ is permitted, is that also true for the Dirac equation?

Answer 1

Bremermann's theorem is derived based on the time-energy uncertainty principle of Quantum Mechanics, and it imposes an upper limit on any physical activity in terms of speed of computation. Fundamentally speaking, what you presented was on point but let's look closely at the doubts you are expressing:

1. Validity of $\Delta E \leq mc^2$: The deduction $\Delta E \leq mc^2$ is born from the rest energy of an object, which is a principal concept supported by relativity theory. It holds a rest mass energy which is equal to $mc^2$, where $m$ is mass and $c$ is the speed of light in a vacuum. In quantum mechanics, energy state is not independent of the state of rest energy and has a pertinent role under the uncertainty principle. Even if there is a joint superposition of states with different levels of energy, the total energy, including the rest energy, should be considered as the uncertainty of the energy.

2. Superpositions with Large Energies: Having a superposition of energy states in different play with energies is certainly a possibility. Nevertheless, this type of estimate of the future state of affairs may be very difficult to determine and it is hard to fully apply in practice. In addition to the uncertainty principle valid for such a state, the uncertainty in energy will be defined in terms of the spread of energies comprised by the quantum superposition.

3. Non-Relativistic vs. Relativistic Considerations: The situation when non-relativistic quantum mechanics applies is not always a case when there are velocities larger than the speed of light although their momentum-energy relationship takes the form of $p > mc$. It leads to the belief that the amount of momentum is greater than the one that is usually discovered by the classical mean. Nevertheless, relativistic effects that are pronounced in the high energies theory, such as reflecting the circumstances, require that approximate use takes place of relativistic quantum mechanics or even the Quantum Field Theory (QFT). The Special Relativity Theory and the associated Special Relativity are far from being violated with such an interpretation in some instances like the Dirac equation where particles can have momenta greater than their mass at rest.

4. Role of Potentials: The creation of a possibility $V \approx mc^2$ may lead to several system behavior changes, such as the particles’ confinement or the emergence of new energy levels. Nevertheless, the instability principle is in effect, and, as a result, this possible state has to be included in the analysis undertaken by the scientists.

According to Bremermann's computation-uncertainty inequality, it is consistent within the quantum mechanics and relativity where the physical process of calculations can have only a specific and limited performance. Albeit there may exist divergent occasions where strange quantum states and relational effects predominate, the fundamentals of the limit are still valid. Besides that, later improvements will into account relativistic effects and gravity, hence they continue the line of work started earlier.