Stability of Matter with RealQM 1

The famous mathematician Elliot Lieb devotes the book Stability of Matter: Quantum Mechanics (311 pages, together with physicist Robert Seiringer) to prove that the total energy $E(\Psi )$ of Standard Quantum Mechanics StdQM in its $3N$-dimensional form for a neutral atom with $N$ electrons, is bounded below for all admissible wave functions $\Psi$. This means that an atom has a ground state as the solution to an eigenvalue problem with smallest eigenvalue as minimal total energy including electronic kinetic energy (positive) and potential energy (negative), and electron-electron repulsion energy adding to stability. 

The case $N=1$ has a direct proof based on a well known Poincaré inequality showing that positive kinetic energy dominates over negative potential energy giving a lower bound to total energy (see below). The rest of the book is devoted to extension to $N>1$ using a Lieb-Thirring inequality and Baxter-Onsager inequality involving a common electron density $\rho (x)$ depending on $x\in R^3$ with $R^3$ 3d Euclidean space. 

Let us check out what RealQM as an alternative to StdQM, has to say about stability of matter. We recall that the wave function $\Psi (x)$ of RealQM depends on $x\in R^3$ as above and has the form of a sum

• $\Psi (x) =\sum_{n=1}^N\psi_n(x)$, 

where the $\psi_n(x)$ are real-valued one-electron wave functions with non-overlapping supports $\Omega_n$ in a decomposition of 3d space, which meet a Bernoulli free boundary with continuity and zero normal derivative and with corresponding charge densities $\rho_n(x)=\psi_n (x)^2$ with unit charge:  

• $\int_{\Omega_n}\rho_ndx = 1$.

The charge density $\rho (x)$ of RealQM is thus simply defined by $\rho (x) =\rho_n(x)$ for $x\in\Omega_n$, which is simply the density of $\Psi (x)$ as function in $H^1(R^3)$ with $\psi_n\in H^1(\Omega_n)$ for $n=1,...,N$ because of the Bernoulli free boundary condition, and $H^1$ is the Sobolev space of functions in $R^3$, which are square integrable along with first derivatives.

It is here interesting to recall what Lieb and Seiringer say on page 31 about the origin of StdQM in 1926: 

• An important historical point is to be noted here. It might have been thought that the correct generalization for N particles is to use N functions of one variable instead of one function of N variables. 
• Such a ‘wrong turn’ did not happen historically, which is, after all, remarkable.

What did not happen was RealQM and so when it now happens 100 years later it may be remarkable. 

The proof of stability of matter for $N=1$ with RealQM directly extends to $N>1$ since its wave function $\Psi\in H^1(R^3) $.

Recall that the proof for $N=1$ starts estimating the potential energy $\int\frac{\psi^2}{\vert x\vert}dx$ using Cauchy's inequality followed by the Poincaré inequality:

•  $\int\frac{\psi^2}{\vert x\vert^2}dx\le 4\int\vert\nabla\psi\vert^2dx$

which follows from the representation 

• $\int\frac{\psi^2}{\vert x\vert^2}dx = -\int 2\psi\nabla\psi\cdot\nabla \ln (\vert x\vert )dx$

using Green's formula and Cauchy's inequality. This proof can be found in my book Applied Mathematics Body and Soul Vol 3, chapter 73. 

PS The first proof of stability of matter within StdQM was given by Dyson and Lenard in 1967 filling over 100 dense pages, which you can read about in the conversation with chatGPT below. The fact that the proof is so lengthy (unreadable) gives yet another indication that StdQM is not a model of physics. If stability of matter, as the most basic physics of the real world, is not clearly present in the model, then the model does not describe real physics. See also next post.

Me: Tell me about the work on stability of matter by Dyson and Lenard.

chatGPT said:

The problem they tackled

In quantum many-body physics you want the ground–state energy $E_{N,K}$ of a system with

• $N$ electrons (charge $-e$) and

• $K$ nuclei with total charge $+Ze$

to obey a linear lower bound

$$E_{N,K}\ge -C(Z)(N+K)(\text{“stability of the second kind”}),$$

i.e. the energy per particle must stay bounded when you add more particles.
Classically this is not obvious – Coulomb attraction grows like $N^{5\mathrm{/}3}$ for a compact cloud – so nothing prevents matter from collapsing. Quantum mechanics gives two tools to fight the collapse:

1. Heisenberg uncertainty principle – localising electrons costs kinetic energy $\sim {\mathrm{\ell }}^{-2}$.

2. Pauli exclusion principle – antisymmetry forces electrons to occupy ever higher momentum states, adding another $N^{5\mathrm{/}3}$‐type repulsion.

The challenge in the 1960 s was to turn these ideas into a rigorous inequality for all $N,K$.

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Dyson & Lenard’s breakthrough (1967–1968)

Paper	Main statement	Key ingredients
F.J. Dyson & A. Lenard, Stability of Matter I, J. Math. Phys. 8 (1967) 423–434	First proof that a non-relativistic Coulomb system is stable provided the negative charges are fermions.	Cell decomposition of space, local uncertainty inequalities, electrostatic screening estimate. AIP Publishing
A. Lenard & F.J. Dyson, Stability of Matter II, J. Math. Phys. 9 (1968) 698–711	Extends the bound to the full neutral system (moving nuclei) and trims technical gaps.	Refined combinatorial counting of cells and careful handling of nucleus–electron correlations. AIP Publishing

Central results

• Linear bound

  $$E_{N,K}\ge -C_{\text{DL}}(N+K),C_{\text{DL}}\approx {10}^{15}$$

  – enormous, but finite.

• Necessity of Fermi statistics
  Dyson also proved that bosonic charges are unstable:
  $E_N\sim -\text{const}{N}^{7\mathrm{/}5}$ — energy becomes super-extensive. Wikipedia

• Introduced the idea that the Pauli principle provides just enough extra kinetic energy ($\int {\rho }^{5\mathrm{/}3}$) to counteract the $\int {\rho }^{4\mathrm{/}3}$ Coulomb attraction.

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How the proof works (bird’s-eye view)

1. Partition space into cubes (“cells”) whose size adapts to the local electron density.

2. Localise each electron in its cube. Heisenberg then gives a kinetic‐energy cost
   $T_{\text{cell}}\gtrsim ({\mathrm{\hslash }}^2\mathrm{/}2m)n_{\text{cell}}^{5\mathrm{/}3}{\mathrm{\ell }}^{-2}$.

3. Estimate Coulomb energy inside and between cells by an electrostatic inequality that replaces the full interaction with a one-body Hartree term plus controllable error.

4. Sum over cells and optimise $\mathrm{\ell }$. The fermionic kinetic term scales like $N^{5\mathrm{/}3}{\mathrm{\ell }}^{-2}$ while the worst negative Coulomb piece scales like $N^{5\mathrm{/}3}{\mathrm{\ell }}^{-1}$. Choosing $\mathrm{\ell }$ proportional to $N^{-1\mathrm{/}3}$ leaves a net contribution that is linear in $N$.

5. Handle nuclei by showing that electrons screen nuclear charge beyond an $O(1)$ distance, preventing additional collapse.

The proof is long (about 100 dense pages over the two papers) because every step has to be made uniform in $N,K$.

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Why it was a milestone

• First complete answer to a question posed by Fisher & Ruelle in 1966 (they famously promised a bottle of champagne for a proof). Wikipedia

• Showed that statistics, not just uncertainty, is essential for the existence of ordinary matter.

• Provided the technical backbone for later, cleaner approaches – most notably Lieb & Thirring’s 1975 proof, which replaced the cell decomposition by the now-famous Lieb–Thirring kinetic-energy inequality and reduced the constant $C$ to something physically reasonable. Wikipedia

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After Dyson & Lenard

• 1975 – Lieb–Thirring: conceptual two-line proof using their spectral inequality; constant $C\sim Z^{7\mathrm{/}3}$.

• 1972-2020: Extensions to relativistic electrons, magnetic fields, QED vacuum polarisation, anyons, etc. (many references in the review literature).

• Comprehensive modern exposition: Lieb & Seiringer, The Stability of Matter in Quantum Mechanics, CUP 2010.

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Bottom line

Dyson and Lenard proved that the simple Hamiltonian

$$H=\sum _{i=1}^N\frac{p_i^2}{2m}-\sum _{i,I}\frac{Z{e}^2}{\mathrm{\mid }{x}_i-R_I\mathrm{\mid }}+\sum _{i<j}\frac{e^2}{\mathrm{\mid }{x}_i-x_j\mathrm{\mid }}+\sum _{I<J}\frac{Z^2{e}^2}{\mathrm{\mid }{R}_I-R_J\mathrm{\mid }}$$

admits a ground-state energy bounded below by $-C(N+K)$ because electrons are fermions. Without that antisymmetry, matter would collapse. Their papers thus supplied the first mathematically rigorous explanation of why tables, rocks and people do not implode under their own electromagnetic forces.