In quantum mechanics and quantum field theory, probabilities are described by mathematical quantities known as probability amplitudes or wave functions. These amplitudes can be extremely small but are typically never exactly zero, at least within the mathematical framework of quantum mechanics.
According to the principles of quantum mechanics, particles and fields can exist in a superposition of states, meaning they can have a nonzero probability of being found in multiple different states simultaneously. However, the probability of finding a particle in a particular state is given by the square of the absolute value of the probability amplitude. While the amplitude itself can be extremely small, the square of a nonzero value is always positive and greater than zero.
That being said, there are certain scenarios in quantum mechanics where certain probability amplitudes can be effectively negligible. For example, in some cases, interference effects between different amplitudes can cause destructive interference, making the net probability in a particular region very close to zero. However, it's important to note that this does not mean the probability is exactly zero but rather that it is extremely small and can be neglected for practical purposes.
In the mathematical formalism of quantum field theory, infinitesimal probabilities are not typically considered as probabilities going exactly to zero. Instead, they are often treated as negligible compared to other probabilities of interest. The concept of infinitesimals in quantum field theory is often encountered in the context of regularization and renormalization techniques, where divergent quantities are manipulated to obtain meaningful results. These techniques involve introducing a cutoff or regulator to handle infinities and obtain finite results, but the probabilities themselves are not regarded as precisely zero.
It's worth noting that the mathematical models and interpretations of quantum mechanics and quantum field theory are subjects of ongoing research and debate among physicists, and different interpretations may have varying perspectives on the exact nature of probabilities and their behavior at extremely small scales.