What is energy?

A concise explanation from Dave Morgan, PhD in Theoretical Physics from William and Mary.

Energy is a mathematical quantity - not a kind of stuff. When physicists talk about energy "transforming from one form into another" they are speaking in a kind of lazy shorthand. I like to make the analogy that energy is like "value". It's a number that we associate with objects. A dollar bill and a quarter both have value. So does an ice cream cone. If you buy a 50 cent ice cream cone with a dollar bill and get two quarters  in change, your dollar didn't "transform" into an ice cream cone. The "value" of your dollar didn't change shape or move somewhere. There was an exchange in the physical world that we can account for by using the abstract quantity we call "value". You exchanged a piece of paper, which has a "value" of 1 for an ice cream cone which has a value of .5 and two shiny metal disks that each have a value of .25.

When physical objects in the universe interact with one another, we can do an accounting of these interactions using an abstract quantity we call "energy". But energy is not any sort of "extra thing" in that physical process.

Also - a great video from astrophysicist Matt O'Dowd (assistant professor at the Physics and Astronomy Department at the Lehman College of the City University of New York.  https://m.youtube.com/watch?v=PUn2izowBkw

What is the difference between Quantum Mechanics and Quantum Field Theory?

An excellent explanation from Barak Shoshany
http://www.perimeterinstitute.ca/people/barak-shoshany

Quantum mechanics is just a set of mathematical tools, a framework, for formulating physical theories. It doesn't tell you anything about particles, or fields, or anything similar. It only tells you about quantum states, amplitudes and probabilities. See In layman's term, what is a quantum state?

Now, you can apply this framework to particles, or you can apply it to fields, or to a variety of other stuff. But every "quantized" theory, including quantum field theory, is built upon the basic principles of quantum mechanics - quantum states, amplitudes and probabilities.

In other words, think of quantum mechanics not as a physical theory by itself, but as a set of basic mathematical rules every self-respecting physical theory must obey.

By the way, quantum field theory itself is also nothing but a framework for formulating physical theories; for example, the standard model of particle physics is the most accurate theory we currently have of forces (except gravity) and matter, and it is formulated using the framework of quantum field theory.

How do we know quantum fields exist?

Excellent explanation by Barak Shoshany, Graduate Student at Perimeter Institute for Theoretical Physics

Fields are a central component in the mathematical formulation of the most successful theories in physics, from electromagnetism to general relativity and quantum field theory. However, scientists have not proven and will never be able to prove that fields, or any other components of our mathematical model, actually "exist" in the same way that tables and chairs "exist".

Quantum field theory is a mathematical framework based on the mathematical concept of a field (a function that has some value at each point in spacetime), together with the mathematical tools of quantum mechanics.

Using this framework we may construct a model of Nature by choosing the types of fields we would like to have and their various properties and interactions. Then we use this model to derive predictions for experimental results.

When we perform these experiments in order to test out theory, we find that the results agree with our predictions to a very high accuracy. Therefore, one can say that scientists have shown that quantum field theory provides a very accurate description of Nature.

In fact, quantum field theory is regarded as one of the most accurate theories of Nature we currently have. So, is it "true"? Do quantum fields actually "exist"? No one knows. Some will tell you that the question itself is meaningless.

Why is that? Well, we wrote some equations and did some calculations, and they turned out to be correct to a certain degree of accuracy. We based our equations and calculations on the premise that fields "exist". However, tomorrow someone else might invent a totally different model, with no fields at all, whose predictions agree with experiment to an even higher accuracy. If we believe that fields exist now, will they suddenly stop existing after that more successful theory is published?

Indeed, many physicists believe that quantum field theory is merely a special case of another, more fundamental theory, that should presumably also include gravity. That theory, which is currently unknown, might or might not include fields as fundamental entities; however, it would have to reduce to quantum field theory at low energies.

Time, Quantum Mechanics and Quantum Fields

Why is time not an operator in Quantum Mechanics even though it is observeable?





One should be careful to distinguish between non relativistic quantum mechanics (Schrödinger equation, etc.) and quantum field theory.

For nonrelativistic quantum mechanics, it is not so surprising that time and space are treated differently, with position being an operator and not time. After all, this is also what happens in Newtonian mechanics: time is absolute, and part of the background, and all other observables are functions of time. This paradigm underlies the formulation of the fundamental problem of Newtonian physics: to determine how a system evolves in time. Time cannot be an observable because an observable is a function of what we consider the system's "state", but the state is considered a function of time in the first place (so time is the independent variable).

Quantum field theory is fully compatible with special relativity, and therefore must treat space and time on equal footing. In nonrelativistic quantum mechanics, position is an observable whereas time is a parameter. That is, position is a function of the state, whereas time is used to label states. In formulating quantum field theory, we therefore have a choice between making spatial coordinates into parameters, or making time into an observable. This choice is discussed by Srednicki:

We can solve our problem, but we must put space and time on an equal footing at the outset. There are two ways to do this. One is to demote position from its stat us as an operator, and render it as an extra label, like time. The other is to promote time to an operator.

Let us discuss the second option first. If time becomes an operator, what do we use as the time parameter in the Schrodinger equation? Happily, in relativistic theories, there is more than one notion of time. We can use the proper time τ$\tau$ of the particle (the time measured by a clock that moves with it) as the time parameter. The coordinate time T$T$ (the time measured by a stationary clock in an inertial frame) is then promoted to an operator. In the Heisenberg picture (where the state of the system is fixed, but the operators are functions of time that obey the classical equations of motion), we would have operators Xμ(τ)$X^{\mu }(\tau )$, where X0=T$X^0=T$. Relativistic quantum mechanics can indeed be developed along these lines, but it is surprisingly complicated to do so. (The many times are the problem; any monotonic function of τ$\tau$ is just as good a candidate as τ$\tau$ itself for the proper time, and this infinite redundancy of descriptions must be understood and accounted for.)

One of the advantages of considering different formalisms is that they may suggest different directions for generalizations. For example, once we have Xμ(τ)$X^{\mu }(\tau )$, why not consider adding some more parameters? Then we would have, for example, Xμ(σ,τ)$X^{\mu }(\sigma ,\tau )$. Classically, this would give us a continuous family of worldlines, what we might call a worldsheet, and so Xμ(σ,τ)$X^{\mu }(\sigma ,\tau )$ would describe a propagating string. This is indeed the starting point for string theory.

Thus, promoting time to an operator is a viable option, but is complicated in practice. Let us then turn to the other option, demoting position to a label. The first question is, label on what? The answer is, on operators. Thus, consider assigning an operator to each point x$\mathbf{x}$ in space; call these operators ϕ(x)$\varphi (\mathbf{x})$. A set of operators like this is called a quantum field. In the Heisenberg picture, the operators are also time dependent:

ϕ(x,t)=eiHt/ℏϕ(x,0)e−iHt/ℏ$\varphi (\mathbf{x},t)=e^{iHt/\hslash }\varphi (\mathbf{x},0)e^{-iHt/\hslash }$.

Thus, both position and (in the Heisenberg picture) time are now labels on operators; neither is itself the eigenvalue of an operator.

So, now we have two different approaches to relativistic quantum theory, approaches that might, in principle, yield different results. This, however, is not the case: it turns out that any relativistic quantum physics that can be treated in one formalism can also be treated in the other. Which we use is a matter of convenience and taste. And, quantum field theory, the formalism in which position and time are both labels on operators, is much more convenient and efficient for most problems.