# The hbar Expansion in Quantum Field Theory

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Referee this paper: arXiv:1009.2313 by Stanley J. Brodsky, Paul Hoyer

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This 2010 paper by S.J. Brodsky and P. Hoyer shows how expansions in powers of Planck's constant $\hbar = h/2\pi$ can give new insights into perturbative and nonperturbative properties of quantum field theories.

''Since $\hbar$ is a fundamental parameter, exact Lorentz invariance and gauge invariance are maintained at each order of the expansion. The physics of the $\hbar$ expansion depends on the scheme; i.e., different expansions are obtained depending on which quantities (momenta, couplings and masses) are assumed to be independent of $\hbar$. We show that if the coupling and mass parameters appearing in the Lagrangian density are taken to be independent of $\hbar$, then each loop in perturbation theory brings a factor of $\hbar$. In the case of quantum electrodynamics, this scheme implies that the classical charge $e$, as well as the fine structure constant are linear in $\hbar$. The connection between the number of loops and factors of  $\hbar$ is more subtle for bound states since the binding energies and bound-state momenta themselves scale with $\hbar$. The  $\hbar$ expansion allows one to identify equal-time relativistic bound states in QED and QCD which are of lowest order in $\hbar$ and transform dynamically under Lorentz boosts. The possibility to use retarded propagators at the Born level gives valence-like wave-functions which implicitly describe the sea constituents of the bound states normally present in its Fock state representation.''

summarized
paper authored Sep 13, 2010 to hep-th
recategorized Aug 27, 2014

When I was a student, the limit $\hbar\to 0$ looked to me like a limit $12 kg\to \infty$. Although it may make sense, the sense is unclear at the beginning. So in reality there are other, non-dimensional, inequalities that describe the corresponding physics in these limits. For example, a quasi-classical approximation in QM corresponds to short-wave approximation when the De Broglie wave length is much smaller than the characteristic length $L$ determining the potential gradient.

Later on, while reading the Akhiezer-Beresteysky "QED", I learned, for example, a non-relativistic approximation, a high-energy approximation, an eikonal approximation, a Born approximation, etc., that employ different inequalities.

Let us see, what we read in the paper abstract: "We show how expansions in powers of Planck's constant hbar = h/2\pi can give new insights into perturbative and nonperturbative properties of quantum field theories."

A good beginning, very promising! I am eager to learn all their insights.

"Since hbar is a fundamental parameter, exact Lorentz invariance and gauge invariance are maintained at each order of the expansion."

A vague statement to me, but let us advance.

"The physics of the hbar expansion depends on the scheme; i.e., different expansions are obtained depending on which quantities (momenta, couplings and masses) are assumed to be independent of hbar."

It's an even more strange statement, because all terms containing $\hbar$ explicitly depend on $\hbar$, the other constants being independent. There is nothing to assume in addition to it, IMHO.

"We show that if the coupling and mass parameters appearing in the Lagrangian density are taken to be independent of hbar, then each loop in perturbation theory brings a factor of hbar. In the case of quantum electrodynamics, this scheme implies that the classical charge e, as well as the fine structure constant are linear in hbar."

If you cannot believe it, read the paper. Under their assumptions, they derive their results and you cannot say the results are wrong. I can point out several assumptions in physics that lead to some unbelievable and incredible results, so what? The results are not guilty since they are correctly derived. Everything is mathematically correct: you take a physical expression, you make your mathematical assumptions and you derive the results. Then you publish them. Easy.

Look at their formula (1) where they make a variable change on the right-hand side $\xi=x/\sqrt{\hbar}$ and do not do it on the left-hand side $A(x_i,x_f ; t_f-t_i)$. They conclude:"Remarkably, the full quantum mechanical structure of the harmonic oscillator model persists as $\hbar\to 0$ when one uses the scaled variables $\xi$".

If you did not get the true meaning of this conclusion, then look at the formula $E_n/\hbar=\omega(n+\frac{1}{2})$ - it is $\hbar$-independent, thus it persists. The inferred insight is: "Thus there is a domain of positions $х\propto\sqrt{\hbar}$ and momenta $m\dot{х}\propto\sqrt{\hbar}$ where the action $S$ is proportional to $\hbar$ and the system stays quantum mechanical even in the $\hbar\to 0$ limit."

Frankly, the ground state is just of this kind, everybody knows it, then how can it stay "quantum mechanical" if it disappears in the limit $\hbar\to 0$? Isn't it a plain delirium?

But don't get upset - their paper contains some correct physical conclusions too. I wish you a happy reading.

Should this not be a (partial) review/answer ...?

No, Dilaton, it is just a warning not to be too superficial while reading their paper.

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