# I Born's rule, causality, and the Dirac equation

#### akhmeteli

Summary
What are the causality properties of the Dirac equation, and how do they relate to Born's rule?
[Moderator's note: Spin off from a previous thread due to topic change.]

Actually, the form of the Hamiltonian does not matter. See Hegerfeldt's paper
Instantaneous spreading and Einstein causality in quantum theory,
Annalen der Physik 7 (1998), 716--725.
Actually, the form of the Hamiltonian does matter. Hegerfeldt admits that his results are not correct for the Dirac Hamiltonian unless one considers only positive energy solutions. And why should we do that? It is clear that a solution of the Dirac equation that vanishes beyond a limited volume (at some time point) hints at the existence of electron-positron pairs, which is something that does seem to happen in nature, so solutions with a superposition of states with positive and negative energies should not be arbitrarily excluded. If you say that this means the Dirac equation is deficient as a one-particle theory, I will agree, but again, why is this the Born rule's fault?

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#### A. Neumaier

Actually, the form of the Hamiltonian does matter. Hegerfeldt admits that his results are not correct for the Dirac Hamiltonian unless one considers only positive energy solutions. And why should we do that?
Because causality together with Lorentz invariance enforce positivity of energy. The Dirac equation is simply defective, and must be patched artificially.
If you say that this means the Dirac equation is deficient as a one-particle theory, I will agree, but again, why is this the Born rule's fault?
The Dirac equation is completely irrelevant for Born's rule.

The fault of Born's rule is that - completely independent of any particular equation - it is not compatible with basic requirements of relativity theory.

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#### N88

Please: Is there a link to an essay making the following point (as a search at the given link did not reveal one):

"The fault of Born's rule is that - completely independent of any particular equation - it is not compatible with basic requirements of relativity theory."

Thank you.

#### A. Neumaier

Is there a link to an essay making the following point: [...] "The fault of Born's rule is that - completely independent of any particular equation - it is not compatible with basic requirements of relativity theory."
Causality together with Lorentz invariance enforce positivity of energy. Hegerfeldt's paper cited in post #690 shows that given positivity of energy, Born's rule implies positive probabilities of a system prepared locally for being observed one second later light years away. This contradicts relativistic causality.

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#### akhmeteli

Because causality together with Lorentz invariance enforce positivity of energy.
Could you explain that?

A. Neumaier said:
The Dirac equation is simply defective, and must be patched artificially.

Yes, it is deficient, but it may be better than any other one-particle equation.

A. Neumaier said:
The Dirac equation is completely irrelevant for Born's rule.

I think it is relevant. In https://arxiv.org/pdf/1902.10778.pdf , Section 3.3, item 4, you describe apparent problems with the Born's rule in the nonrelativistic case (as far as I can judge by footnote 16 there). There are no such problems in the case of the Dirac equation. As far as I can see, you never explained how item 5 there (about K/P) is relevant/applicable).

A. Neumaier said:
The fault of Born's rule is that - completely independent of any particular equation - it is not compatible with basic requirements of relativity theory.
It will be easier to understand your reasoning after you explain why "
causality together with Lorentz invariance enforce positivity of energy". And I do object to "
completely independent of any particular equation" - you have yet to show a problem with the Born's rule in the case of the Dirac equation.

#### A. Neumaier

Causality together with Lorentz invariance enforce positivity of energy.
Lorentz invariance enforces unitary representations of the Poincare group. These fall into direct sums of irreducible representations, classified by Wigner. Only the irreducible representations of nonnegative mass and timelile or lightlike momenta are causal.
And I do object to "completely independent of any particular equation" - you have yet to show a problem with the Born's rule in the case of the Dirac equation.
For the failure of Born's rule in the relativistic case it is irrelevant that the Born rule is consistent with an unphysical equation that allows particles to move into the backward light cone.
As far as I can see, you never explained how item 5 there (about K/P) is relevant/applicable).
I replaced it by the more general reference to Hegerfeldt. The Keizer/Polyzou models all have positive energy. That's why they are practically useful.

#### akhmeteli

Lorentz invariance enforces unitary representations of the Poincare group. These fall into direct sums of irreducible representations, classified by Wigner. Only the irreducible representations of nonnegative mass and timelile or lightlike momenta are causal.
My understanding is the Dirac equation is Lorentz invariant and causal, but not energy-positive. Timelike momentum does not necessarily mean positive energy (http://mathworld.wolfram.com/Timelike.html).
A. Neumaier said:
For the failure of Born's rule in the relativistic case it is irrelevant that the Born rule is consistent with an unphysical equation that allows particles to move into the backward light cone.

Again, my understanding is the Dirac equation is causal. Maybe I am missing something, but I believe "movement into the backward light cone" just means that the Dirac equation can describe the past as well. The Dirac equation is indeed deficient as a one-particle equation, but again, it is better than any other one-particle equation.

A. Neumaier said:
I replaced it by the more general reference to Hegerfeldt.
Where did you replace it? And however general Hegerfeldt might be, his results are not applicable to the Dirac equation.
A. Neumaier said:
The Keizer/Polyzou models all have positive energy. That's why they are practically useful.
And the Dirac equation is not practically useful? With all due respect, this is rich.

#### A. Neumaier

My understanding is the Dirac equation is Lorentz invariant and causal, but not energy-positive.
Your understanding is faulty., as the Dirac equation has acausal solutions.

#### akhmeteli

Your understanding is faulty., as the Dirac equation has acausal solutions.
Could you please provide a reference?

#### A. Neumaier

Could you please provide a reference?
This was already known to Dirac, who had to discard half of the solutions. Look at his first paper. Only the postive energy part has a physical meaning, and that part is covered by Hegerfeldt's argument.

#### akhmeteli

This was already known to Dirac, who had to discard half of the solutions. Look at his first paper. Only the postive energy part has a physical meaning, and that part is covered by Hegerfeldt's argument.
I assume that "Dirac's first paper" is Proc. Roy. Soc., A, Vol. 117, pp. 610-624 (1028). I failed to find there anything about acausal solutions of the Dirac equation. I don't understand why the negative energy part does not have a physical meaning - it describes positrons (for example, as holes). Again, the Dirac equation is deficient as a one-particle equation, but why is it not causal? Until you provide a proper reference, I reject your statement "the Dirac equation has acausal solutions".

#### A. Neumaier

I assume that "Dirac's first paper" is Proc. Roy. Soc., A, Vol. 117, pp. 610-624 (1028). I failed to find there anything about acausal solutions of the Dirac equation. I don't understand why the negative energy part does not have a physical meaning - it describes positrons (for example, as holes). Again, the Dirac equation is deficient as a one-particle equation, but why is it not causal? Until you provide a proper reference, I reject your statement "the Dirac equation has acausal solutions".
It describes a positron only after applying time reversal. But time reversal makes the electrons behave acausal. To have a causal interpretation one needs to reverse half of the eigensolutions. This violates the superposition principle. A linear combination of an electronic solution and a positroni c solution of the Dirac equation has no causal interpretation at all.

#### akhmeteli

It describes a positron only after applying time reversal. But time reversal makes the electrons behave acausal. To have a causal interpretation one needs to reverse half of the eigensolutions. This violates the superposition principle. A linear combination of an electronic solution and a positroni c solution of the Dirac equation has no causal interpretation at all.
This can only suggest that the Dirac equation is deficient as a one-particle equation. This does not change the fact that the Dirac equation is causal and does not have acausal solutions, unless you use your very own definition of causality. Do you really dispute that a wave function that initially has a compact support evolves within the light cone under the Dirac equation? If you don't think this is a mathematical fact, why don't you provide a reference?

#### A. Neumaier

This can only suggest that the Dirac equation is deficient as a one-particle equation. This does not change the fact that the Dirac equation is causal and does not have acausal solutions, unless you use your very own definition of causality. Do you really dispute that a wave function that initially has a compact support evolves within the light cone under the Dirac equation? If you don't think this is a mathematical fact, why don't you provide a reference?
Equations that are deficient as 1-particle equations are unphysical. Causal (local support) solutions of the Dirac equations have no physical interpretation as they superpose states propagating forward and backward in time, and hence have no causal meaning.

#### vanhees71

Gold Member
That said, it's also important to emphasize that relativistic microcausal (aka local) QFT is a very successful formulation of relativistic QT. There's no need of oldfashioned single-particle interpreted equations anymore. The Dirac equation is nowadays simply contained in the general QFT framework (particularly in terms of modern QED, which is equivalent to Dirac's hole theory, which is only too cumbersome to work with instead, and it also shows that the single-particle interpretation is impossible, but relativistic physics enforces a many-body theory right at the foundations).

For the free Dirac fermion, it's precisely what Arnold said, but applied to field operators rather than first-quantization interpreted wave functions. The free-Dirac-fermion operator is
$$\hat{\psi}(x)=\sum_{s=\pm 1/2} \int_{\mathbb{R}^3} \mathrm{d}^3 p\frac{1}{\sqrt{(2 \pi) ^3 2 E_p}}[\hat{c}(\vec{p},s) \exp(-\mathrm{i} p \cdot x) + \hat{d}^{\dagger}(\vec{p},s) \exp(+\mathrm{i} p \cdot x)]_{p^0=E_p}.$$
So there are field modes with positive and negative frequencies, but the QFT formalism alows one to write annihilation operators in front of the former and creation operators in front of the latter. Using the Poincare-transformation properties of the Dirac spinor one can show that this field operator leads to a positive semidefinite energy spectrum (existence of the stable ground state, called "vacuum") provided one quantizes the field as fermions, i.e., with annihilation and creation operators fulfilling fermionic canonical anti-commutation rules. At the same time the theory is causal, i.e., the microcausality principle is fulfilled, which means that for two local observable operators (all of which can be built with help of the Dirac-fermion field operators) commute at space-like distances, i.e.,
$$[\hat{O}(x),\hat{O}'(y)]=0 \quad \text{for} \quad (x-y)^2=(x^0-y^0)^2-(\vec{x}-\vec{y})^2<0.$$
This is the modern version of the Feynman-Stückelberg trick, and nothing runs backward in time by construction, and that's the reason for writing a creation operator in front of the negative-fequency field mode. You simply describe not only one sort of particles by this quantized Dirac equation but a particle and its antiparticle, which necessarily have the same mass but opposite charge (where charge is the Noether charge of the invariance under multiplication of the field operator with an arbitrary phase; gauging this symmetry leads without further ado to QED).

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#### akhmeteli

Equations that are deficient as 1-particle equations are unphysical.
So the Dirac equation is unphysical. I suspect you use some innovative definition of unphysical.
A. Neumaier said:
Causal (local support) solutions of the Dirac equations have no physical interpretation as they superpose states propagating forward and backward in time, and hence have no causal meaning.
I object to what looks like an attempt to redefine causality on the fly: causal does not necessarily mean "local support": all solutions of the Dirac equation are causal. There are no acausal solutions of the Dirac equation. If you disagree, please give a reference. This is not the first time I request such a reference. You choose to ignore such requests.

#### vanhees71

Gold Member
The 1st-quantized Dirac equation has no consistent physical interpretation (at least not in the case of interacting particles or a single particle interacting with an external field). This lead Dirac to his "hole" theory, which is a many-body reinterpretation. Since hole theory is utmost complicated, today one prefers the use of the 2nd-quantization formalism (i.e., quantum field theory).

#### akhmeteli

The 1st-quantized Dirac equation has no consistent physical interpretation (at least not in the case of interacting particles or a single particle interacting with an external field). This lead Dirac to his "hole" theory, which is a many-body reinterpretation. Since hole theory is utmost complicated, today one prefers the use of the 2nd-quantization formalism (i.e., quantum field theory).
I agree, and I repeatedly admitted that the Dirac equation is deficient as a one-particle equation. Let me, however, remind readers of this thread the context of my discussion with @A. Neumaier.

In the first version of his preprint https://arxiv.org/pdf/1902.10778v1.pdf, A. Neumaier offered, in particular, the following critique of the Born's rule:

"4. When a particle has been prepared in an ion trap (and hence is there with certainty), Born’s rule implies a tiny but positive probability that at an arbitrarily short time afterwards it is detected a light year away.17... Thus $|\psi(x)|^2$ cannot be the exact probability density for being detected at $x$.
5. This argument against the exact probability density interpretation of $|\psi(x)|^2$ works even relativistically, due to the existence of the Newton-Wigner position operator for massive particles.

[footnote]17 Indeed, for a single massive particle, Born’s rule states that $|\psi(x,t)|^2$ is the probability density for locating at a given time $t$ the particle at a particular position $x$ anywhere in the universe, and the Fourier transform $|\tilde{\psi(p,t)}|^2$ is the probability density for locating at a given time $t$ the particle with a particular momentum $p$. In the present case, the position density has bounded support, so by a basic theorem of harmonic analysis, the momentum density must have unbounded support. This implies the claim."

(In the second version of his preprint https://arxiv.org/pdf/1902.10778.pdf he replaced item 5.)

This critique seemed strange to me (although I am no fan of the Born's rule), so I wrote (https://www.physicsforums.com/threads/the-thermal-interpretation-of-quantum-physics.967116/post-6141961): "I guess this is only true if one assumes nonrelativistic equation of motion? (The argument about Newton-Wigner does not seem clear)."

A. Neumaier replied (https://www.physicsforums.com/threads/the-thermal-interpretation-of-quantum-physics.967116/post-6141964), saying that there is no consistent relativistic particle picture in quantum mechanics, except in the free case. And he stated that "...for a free particle, if one would know the position at one time to be located in a small compact region of space, it could be the next moment almost everywhere with a nonzero probability."

As this is not the case for the Dirac equation, I could not accept this statement, and this is how this can of worms was open:-) So my main argument was: OK, if the Dirac equation is not good enough, how is this the Born's rule's problem?

"Born's rule, causality, and the Dirac equation"

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