Where does the energy come from that produces virtual particles?

In ordinary Newtonian physics, just about everything can be traced back to some elementary process that conserves energy and momentum. For 400 years we were taught that neither energy nor mass could be created or destroyed but had to be conserved througout some process such as the moon orbiting earth. We also learned that conservation laws applied to closed systems that you could see and systems that you could not see, from the cosmic to the atomic. Does a tree falling in the forest when no one is there to see it, still conserve energy? you bet! But then during the earth-20th century, the Roaring Twenties hit, and physics was turned upside down for a few years.

I am not going to review quantum mechanics and quantum field theory in this writing because you have probably read most of the literature about this 'Second Pillar' of physics. The important thing to remember is in the atomic world, a whole new set of paradighms apply that have nothing to do with Newton's Physics except in some skelletal form. We still talk about mass, momentum and energy, but now the objects of our concern are elementary objects that behave as waves or particles depending on what kinds of experiment you put them through. Energy is no longer a Newtonian quantity but is an 'operator' that acts on a particle wavefunction to return a value for a particular state index. Momentum also has its own operator, and the way these operators act on a wavefunction is analogous to how a particular tuning fork vibrates in resonance to an applied force. Each vibratory mode of the wavefunction of an electron has its own energy at a particular instant in time, and a particular momentum at a particular position in space. Physicists say that energy and time 'commute' with each other and momentum and position do likewise. Because these wavefunctions are statistical in nature, the 'square' of a component of the wavefunction gives the probability that the electron will have a specific energy and momentum. But this statistical feature of an electron's state means that the product of the conjugate variables must be greater than or equal to Planck's constant. This gives us the famous Heisenberg Uncertainty Principle:

What these relations relate to is our ability to distinguish between each of the possible energy and momentum quantum states of an electron at a particular moment in time, and a particular location in space. In fact, because we are dealing with states that are part of an infinite harmonic series for the electrons wavefunction, we can use the mathematics of Fourier to relate frequency to wavelength for each of the states. In light and sound we have wavelength = constant / frequency where the constant is the speed of sound or light. In quantum mechanics, the wavefunction is based on similar relationships for the conjugate variables (E,t) and (p,x). The experimental problem is that because E and t are conjugate, it means that as we try to specify the momentum state, p, more accurately we steadily lose accuracy in knowing where the electron is in the x variable. Similarly, as we try to precisely determine how much energy a system has, E, we lose accuracy in knowing at which specific instant it had that energy.

What does this have to do with the energy of virtual particles?

The Heisenberg relatonship between energy and time is actually a statement of how well we can know both of these quantities for any system that has wavelike properties. In words:

The uncertainty in the total energy of a particular state decreases as the amount of time it is in that state increases.

This is often interpreted as a statement of our being able to measure the energy of the system if we only observe it for a short while. A practical example is as follows.

Initially at Time = Ti our system consists of two particles Pa and Pb which have the total energies of Ea and Eb. Then Einitial = Ea + Eb. A neighboring state at time =T2 contains the same two particles and their energies, but includes a third particle V, with the energy Ev. The final state of the system at time = Tf contains only the original two particles. According to Heisenbergs Uncertainty Principle, the change in energy between the two states is just (Ea + Eb + Ev) - (Ea + Eb) = Ev. This energy change between the two states is related to how long then state with the third particle exists according to Delta-T = h/Ev which is the minimum time the Ev energy can persist.

In quantum mechanics a system begins in an initial state at Time Ti and ends in a final state at Tf. These states contain only the original particles, in this case A and B. What happens in-between can include any other process so long as it obeys Heisenbergs Uncertainty Principle so that

Ev = h/(Tf-Ti)

If the time between the initial and final states is long, the energy fluctuation, Ev, will be very small, but if the time difference is short, the value for Ev can be very large.

So where does this energy Ev come from? You can think of it as being 'borrowed' from the state in which the particle V did not exist...which is called the quantum vacuum. That's because the vacuum state is the lowest energy state of the system remaining after we remove the two original particles. What is left over is n 'empty space' in which all of the other energy fluctuations ( interpreted as virtual particles because of E=mc^2) that come and go over time periods set by the amount of energy they contain.

Another way to think of this is to use the measurement analogy for what happens when you average together lots of measurements. When you start out with 36 measurements and average them, you get an answer but this is the mid-point of a bellcurve for these repeated measurements that has a 'standard deviation', which tells you the spread of the measured points around the average value. As you increase the number of measurements to 10,000 your average may not change by much, but now the shape of the bellcurve has narrowed because the standard deviation is now squareroot(10000/36) = 100/6 times smaller. The more you measure the smaller becomes the fluctuation in the parameter you are measuring. In the same way, you make 36 energy measurements of a particle state and the standard deviation is determined by Heisenbergs Uncertainty principle based on the amount of time involved in making these measurements. But when you make more measurements you increase the time between Ti and Tf and the standard deviation decreases to a smaller value.

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