Newtonian Forces

Force is a fundamental concept present in any physics curriculum and is central to classical mechanics. However, this idea is often misapplied within classical physics, as will be demonstrated below.

Forces are introduced through Newton's mechanics, typically via the second law, expressed as F=ma. While I will critique this formulation in a future post, for now, let's accept it as the accurate representation of the law. By definition, forces are vectors characterized by both magnitude and direction. This is because acceleration ( a ) is also a vector, and multiplying a vector by a scalar ( m ) yields another vector. In other words, forces and accelerations are intimately connected.

In electromagnetism, the concept of force remains applicable, as charged particles experience the Lorentz force, which accelerates them and alters their velocity, whether in magnitude or direction.

However, complications arise when we examine gaseous systems. To compress a gas, a force must be exerted on its container; yet, this application does not yield acceleration, as it is nonsensical to discuss a gas's velocity (refer to my previous post). To address this dilemma, physicists invented the concept of pressure, defined as the force component acting perpendicularly on a surface, normalized by the area of that surface. It is essential to note that pressure is a scalar, while force is a vector from which we extract its magnitude along a specific direction.

As forces are contingent upon accelerations, another complication arises when measuring in accelerated (non-inertial) frames of reference. In such frames, forces termed "fictitious" or "apparent" emerge, defined as ( F = -ma ), where ( a ) represents the frame's acceleration (in this instance, this expression holds true, unlike in Newton's second law). These fictitious forces are as real as other familiar forces, such as gravity or elasticity. Consequently, many students struggle to differentiate between centrifugal and centripetal forces.

What distinguishes fictitious forces from real ones? The key difference lies in the fact that real forces arise from interactions. Gravity manifests when a mass interacts with another mass. The elastic force occurs when a mass interacts with a spring or a similar system. The centripetal force exists only when there is an interaction between an object and the mechanism maintaining its circular path (e.g., a ball tethered to a rope experiences centripetal acceleration due to its interaction with the rope; cutting the rope removes that interaction, and thus, the force).

Fictitious forces exist solely in non-inertial reference frames and lack a source; they are merely a byproduct of how we conduct measurements. In a reference frame where the ball mentioned earlier is stationary, we must "pretend" a force is acting on it to negate its acceleration in that frame. The fictitious force must counteract the only real force: the centripetal force exerted by the rope. When a vehicle skids around a curve, the cause is not a centrifugal force; rather, it is the absence of the centripetal force (the static friction between the tires and the road). The deviation from the intended path is a consequence of Newton's first law, as observed in an inertial frame.

Non-Newtonian Forces

Until the early 20th century, the terms "force" and "interaction" were nearly interchangeable. Every interaction produced forces, and every non-fictitious force resulted from an interaction. However, the discovery of beta decay has changed this understanding, even though many textbooks still refer to the "weak force."

Beta decay involves the transformation of a neutron into a proton, electron, and neutrino. This process can also transform a proton in a nucleus into a neutron, positron, and neutrino, though this reaction is limited to nuclei due to energy conservation constraints.

Such a phenomenon cannot be explained by any of the known fundamental forces—neither gravitational nor electromagnetic. For various reasons, the strong nuclear force, which binds protons and neutrons within nuclei, cannot account for beta decay. Enrico Fermi first theorized beta decay, introducing the concept of weak interaction, which is often regarded as the fourth fundamental force.

Most textbooks assert that there are four fundamental forces in the universe: the strong force, weak force, electromagnetic force, and gravitational force. This statement is rather misleading. Most people associate a force with something that either pushes or pulls. It seems natural to link a vector to any push or pull. However, in processes involving the weak force, no such push or pull exists; there is no location to which a vector can be assigned.

In fact, the weak force should not be classified as a force, as forces can be represented by vectors, and in decay processes, there is no point to which a vector can be assigned. Instead, it is an interaction. Not all forces result from interactions (e.g., apparent forces), and not all interactions produce a force (e.g., weak interactions). Forces can be connected to interactions that effectuate a push or pull on an object. Therefore, they cannot be responsible for quantum mechanical phenomena in which position lacks significance (as discussed in episode 1).

What constitutes interactions? They can be defined as any influence that alters the state of a system. For instance, consider a particle in free motion. When it is within Earth's gravitational field, its interaction with this field produces a force that changes its velocity by adding a vertical component. Consequently, the particle's state changes (noting that if we specify the initial state of the particle in a reference frame where it was at rest at ( t=0 ), its state remains unchanged in the absence of a gravitational field).

A gas can interact with a thermal reservoir, exchanging heat without any forces involved. Nevertheless, the state of the gas can still change, affecting its pressure, volume, and/or temperature. The state of a gas can be visualized as a point on a pressure-volume diagram, with its transformation represented as a curve on the ( pV )-plane. This abstract representation of state and interactions is not exclusive to quantum mechanics.

A neutron can engage with the weak interaction field, transitioning from a state of rest to a state containing a proton, electron, and neutrino, with energy and momentum such that the total final momentum is zero, and the total energy equals the neutron's mass. No force is involved, yet a state transformation occurs.

The weak field originates from the neutron itself, as it continuously emits quanta of the weak field. The neutron's interaction with these quanta can lead to state transformation, akin to how an electron interacts with its own electric field. Even classically, an electron should interact with the electric field generated by itself, although this issue is often overlooked. Despite leading to an infinitely strong coupling (since the intensity of the field decreases as ( 1/r^2 )), it does not yield any measurable effects.

In contemporary beta decay theory, a neutron (specifically one of its constituent quarks) emits a charged field, losing a negative charge and transforming into a proton. This field materializes into an electron (which carries the electric charge) and a neutrino (which carries the necessary momentum and angular momentum to conserve it).

The next episode will delve into the nature of electrons: Are they particles or waves? Join us to explore this question (though I do not presume to have a definitive answer, I believe I can help clarify the concept of an electron more effectively than is usually presented).

The first video, Demystifying Quantum Mechanics | Chad Orzel, offers insights into the complexities of quantum mechanics and how to make it more accessible.

The second video, FLAVIO DEL SANTO: Towards a Measurement Theory in QFT, discusses the foundations of measurement theory within quantum field theory, shedding light on its implications for our understanding of forces and interactions.