Let's first address the elephant in the room. Ayn Rand was a philosopher who would not engage in any philosophical debate with claims that ran contrary to her common-sense vision of the world. A theory or a philosophical stance which attempted refuting Rand's claims would typically be ridiculed as "irrational", "bad science", or "bad philosophy". Most Objectivist followers today, who accept Leonard Peikoff as Rand's heir, follow suit in almost all of Rand/Peikoff claims. This includes a vehement (and bizarre) misinterpretation of Kant as a philosopher who was anti-Enlightenment. Because of this particular attitude shown by many Objectivists, they are not engaged with by the academia, not to mention members of scientific community.
With that out of the door, Ayn Rand's account of causality (and free will) is incompatible with modern physics. By Rand's account, each object has an identity and its causal powers. However, since objects are field excitations sustained by various mechanisms of repellment, they do not exist in the solid "object" sense on the particle level. They are also driven by forces, laws and interactions beyond their control, and as such, they cannot possess independent causal powers. And no modern physics theory would posit such causal powers. Not even Bohmian mechanics that was already mentioned here would work as it implies a rather extreme type of determinism which has no place for free will. The view that a stone strives to reach its natural goodness by gravitating towards earth is Aristotlean essentialism:
how does Objectivism reconcile free will with causality? The answer to
that question is that Objectivism has an Aristotelian,
entity-causation view of causality. Causality is the law of identity
applied to actions, and actions are actions of entities. It is
entities, i.e., things, that act. A ball rolls, an atom decays, etc.
Thus, Rand has to reject large portions of modern physics and retort to classical mechanics coupled with Aristotleanism to retain her stance. And this is indeed the case.
Objectivists like Leonard Peikoff, David Harriman, and Petr Beckmann reject modern physics deeming it Kantian, therefore "bad", and wish us to go back to Newtonian physics where time is absolute (See: The Philosophic Corruption of Physics). Naturally, they have to reject quantum mechanics for its probabilistic nature and underlying mathematic abstraction along with Einstein's theory of relativity, for similar reasons.
Historically speaking, in the 80s, Objectivist Petr Beckmann, an associate of Ayn Rand, led an anti-Einsteinian physics journal Galilean Electrodynamics. Beckmann also wrote a book Einstein plus Two that aimed to "debunk" Einstein's relativity. More modern Objectivist publications of Harriman and Peikoff also attempt put quantum mechanics and Einstein's theory into question.
Thankfully, those are investigated by Warren C. Gibson as a part of the Ayn Rand Studies series. Gibson engages with those in Modern Physics versus Objectivism paper2. From the Abstract:
Leonard Peikoff and David Harriman have denounced modern physics as
incompatible with Objectivist metaphysics and epistemology. Physics,
they say, must return to a Newtonian viewpoint; much of relativity
theory must go, along with essentially all of quantum mechanics,
string theory, and modern cosmology. In their insistence on
justifications in terms of “physical nature,” they cling to a
macroscopic worldview that doesn’t work in the high-velocity arena of
relativity or the subatomic level of quantum mechanics. It is
suggested that the concept of identity be widened to accommodate the
probabilistic nature of quantum phenomena.
Warren C. Gibson, having substantial knowledge in the field of physics, engages and quotes numerous passages from various books produced by two prominent Objectivists. In the paper itself, there are a plethora of remarks (along with equations) against the Objectivist stance on modern physics. Hence, I am only going to bring in some excerpts. For a longer read and the argument, I encourage getting and reading the paper itself.
To summarize the point, briefly, Gibson states:
In each area, we find similar complaints from P&H [Peikoff & Harriman]:
(1) physicists fail to provide “physical explanations” for their theories;
(2) they are on the wrong track because they fail to follow Rand’s epistemological prescriptions; and
(3) although it is generally not stated, they imply
that physicists should mend their ways and get with the
Generally, both Peikoff and Harriman deem that modern physics makes the external world "noumenal" (i.e. "Kantian/Evil"), leading to a form of "mysticism". Take quantum uncertainty, for that matter. Quantum uncertainty drives Objectivists "crazy". Thus, they reject the probabilistic nature of, and the existence of wave function itself. Gibson:
we see them insisting that nature must conform to their macroscopic
visions of how the world ought to be—essentially, what Ayn Rand could
have seen or visualized in her living room. But if they wanted to
preserve the concept of identity, philosophers could simply recognize
that probability is part of the nature of a radioactive atom— part of
its identity. Then the laws of identity and causality would be
Today, we understand that entities such as point particles are merely mathematical abstractions and lack spatiality. Yet, we see Peikoff and Handimann rejecting such notions as "fantastic":
For Newton, mathematical equations are valid because they describe
physical facts learned from observation. For Einstein, mathematical
equations are valid because they belong to a certain self-contained, a
priori system of numbers (Peikoff)
They disapprove of any mathematical-physical abstractions that “cannot be interpreted physically”(Peikoff). Futhermore, Peikoff states that:
The physical world, in this new approach, is not a primary, but a derivative of a non-physical realm consisting of pure mathematics. (Peikoff)
So, for Peikoff, it is wrong if “concepts are logically prior to percepts”(Peikoff) because then we operate in the realm of "fantastic premises".
It does indeed resemble a caricature view of Mach's positivism, a little. Gibson concludes:
We have seen P&H repeatedly calling for physical explanations of
scientific theories. They seem to be insisting on explanations that
conform to Objectivist ontology, and if the findings of physics fail
to conform to their demands, then those findings must be tossed. They
demand that physics return to the worldview of Newton, where objects
interact with other objects strictly according to the nature of those
objects. An apt metaphor would be billiard balls colliding on a
billiard table. P&H want a world where particles have definite
attributes that determine their behavior. They want a world where the
“nature” of objects can be determined in isolation from their
environments and from the observers that comprehend them.
Thus, the remedy would involve:
[C]oncepts like object, nature, and cause as they appear in
Objectivism need to be expanded or revised. Perhaps the concept of the
“nature” of an object can be extended to include probability as in
quantum mechanics. Perhaps causality needs amendment or extension.
Perhaps the uncertainty principle can be reconciled with causality.
Objectivists need to confront the aforementioned conflict between
Newtonian determinism, where P&H would take us, and free will. They
need to ask themselves whether they are clinging stubbornly to
macroscopic visualizations that must be abandoned if microscopic
phenomena are to be comprehended.
As it currently stands, no wonder that no physicist would engage in such "debate" with Objectivists, especially that any such debate has been notoriously impossible due to the character of Rand's philosophical heir - L. Peikoff.
Note also that Warren C. Gibson's study was published within means that don't belong to Ayn Rand Institute. The Journal of Ayn Rand Studies like The Atlas Society are not organizations affiliated with The Ayn Rand Institute. Those organizations distance themselves from Ayn Rand Institute for its dogmatism and lack of open mindness. So, it might not be the case that all Objectivists share the claims made by L. Peikoff.