I'm no physics historian, but it can be seen that one can "follow the math" to discover if there are useful physical principles that result. An example is Max Planck who, when wrangling with the "Ultraviolet Catastrophe" of black body radiation, tested various mathematical models to see which fit the experimental data. The result was a mathematical relationship that indicated energy might be quantized in some physical cases, and not always be continuous in nature. That was a useful path, and a productive one for both science and engineering.
The demand that negative probabilities exist -- demanding that what should be an absolute value quantity is negative, appears to be torturing the mathematics rather than following it. As a thought experiment, it led Dirac to the negative energy sea of virtual particles. But, like virtual particles, is the negative probability model just as "virtual?" Fitting Reality to your preferred formula has become more prevalent with time. We are likely experiencing more and more dead-ends in science because of this desire to conform to the "pretty" mathematics rather than conforming to "ugly" experimental data.
Just for kicks, here is an example of a ghost solution from middle school.
A boat travels from point A to point B and back. The speed of the current is R, the velocity of the boat without current is V, and its average velocity for the whole round-trip is U. Assume that the speed of the boat is V + R when it travels with the current, and V - R when it travels against the current. Then it is easy to see (math!!) that the three quantities V, U, and R are connected by the equation V^2 - VU - R^2 = 0.
Let R=2, and U=3. Then solving for V we get 2 solutions: 4 and -1. That -1 is a "ghost solutions" - it does not have a physical interpretation.
In the case of throwing an object described in that thread, negative time ackchyually can be made sense of:
The equation S = (at^2)/2 + ut he brings is - in the general case - that of a body acted upon by a constant force. The full(er) equation is S = (a(t-t0)^2)/2 + u[t0](t-t0) + S[t0], where t0 is some time point we have selected as a starting point. Thus it's the difference t-t0 that can take a negative value, which just means that t is a point in time before t0.
The problem here is that the model does not adequately describe our situation where the force is not constant - the object was thrown and it will hit the ground, violating that assumption.
I can't really comment on his other examples, since my knowledge of physics does not extend significantly beyond waves and oscillations, and the special theory of relativity.
Feynman, like all modelers, falls to the classic hazard…confirmation bias. So, he sidesteps the "innards" of the math because they might throw a wrench into his model!
If probability is zero, then the event never occurs. If the probability is negative, then it REALLY never occurs, in fact it never occurs in another universe.
Unless I'm severely mistaken probabilities are usually interpreted as statements about future outcomes. How come there is no distinction between these and probabilities of the past? In the sense that don't we also come to derive conclusions based on probability of things we didn't observe but have already happened?
Could that be an interpretation of a negative probability? It's the same as a positive except it's already been. We're now attempting to make a statement about how that came to be. So in an attempt to figure out exactly what might have happened we now have to resort to working with a negative probability.
I'm no physics historian, but it can be seen that one can "follow the math" to discover if there are useful physical principles that result. An example is Max Planck who, when wrangling with the "Ultraviolet Catastrophe" of black body radiation, tested various mathematical models to see which fit the experimental data. The result was a mathematical relationship that indicated energy might be quantized in some physical cases, and not always be continuous in nature. That was a useful path, and a productive one for both science and engineering.
The demand that negative probabilities exist -- demanding that what should be an absolute value quantity is negative, appears to be torturing the mathematics rather than following it. As a thought experiment, it led Dirac to the negative energy sea of virtual particles. But, like virtual particles, is the negative probability model just as "virtual?" Fitting Reality to your preferred formula has become more prevalent with time. We are likely experiencing more and more dead-ends in science because of this desire to conform to the "pretty" mathematics rather than conforming to "ugly" experimental data.
Just for kicks, here is an example of a ghost solution from middle school.
A boat travels from point A to point B and back. The speed of the current is R, the velocity of the boat without current is V, and its average velocity for the whole round-trip is U. Assume that the speed of the boat is V + R when it travels with the current, and V - R when it travels against the current. Then it is easy to see (math!!) that the three quantities V, U, and R are connected by the equation V^2 - VU - R^2 = 0.
Let R=2, and U=3. Then solving for V we get 2 solutions: 4 and -1. That -1 is a "ghost solutions" - it does not have a physical interpretation.
I knew I had seen something like your comments recently.
See this for more examples:
https://x.com/getjonwithit/status/1865878619588817246
In the case of throwing an object described in that thread, negative time ackchyually can be made sense of:
The equation S = (at^2)/2 + ut he brings is - in the general case - that of a body acted upon by a constant force. The full(er) equation is S = (a(t-t0)^2)/2 + u[t0](t-t0) + S[t0], where t0 is some time point we have selected as a starting point. Thus it's the difference t-t0 that can take a negative value, which just means that t is a point in time before t0.
The problem here is that the model does not adequately describe our situation where the force is not constant - the object was thrown and it will hit the ground, violating that assumption.
I can't really comment on his other examples, since my knowledge of physics does not extend significantly beyond waves and oscillations, and the special theory of relativity.
Feynman, like all modelers, falls to the classic hazard…confirmation bias. So, he sidesteps the "innards" of the math because they might throw a wrench into his model!
In finance, the breakdown of models is the definition of crisis.
If probability is zero, then the event never occurs. If the probability is negative, then it REALLY never occurs, in fact it never occurs in another universe.
Unless I'm severely mistaken probabilities are usually interpreted as statements about future outcomes. How come there is no distinction between these and probabilities of the past? In the sense that don't we also come to derive conclusions based on probability of things we didn't observe but have already happened?
Could that be an interpretation of a negative probability? It's the same as a positive except it's already been. We're now attempting to make a statement about how that came to be. So in an attempt to figure out exactly what might have happened we now have to resort to working with a negative probability.
Good guess, but probability is with respect to any proposition, which can be for any time, given certain evidence you assume. Simple as that.