Heisenberg's uncertainty principle
I keep seeing Heisenberg's uncertainty principle described in popular journalese as allowing a temporary violation of the law of energy conservation (or of the law of momentum conservation). The argument goes that HUP allows you to lend or borrow energy as long as settlement is made very soon, and that this arrangement represents a temporary violation of the law of energy conservation.
The truth is that there is no violation of the law of energy conservation.
The lending or borrowing of energy (and momentum) is done in a way that always respects energy (and momentum) conservation. How can this be? Just as with financial transactions where you lend to someone or borrow from someone, with energy transactions you lend to something or borrow from something. What is that something? The details depend on the precise circumstances, but I can illustrate one of the possibilities with the diagrams below.
Walk through these diagrams:
- A: This shows a particle going along all by itself conserving energy (and momentum) as per usual.
- B: This shows a particle that goes along as in A, but then it emits another type of particle (shown as the dashed line), after which the energy (and momentum) of the original particle have changed. Call these particles of type 1 (solid line) and type 2 (dashed line).
- C: This shows particles of types 1 and 2 going along, but then the type 2 particle is absorbed by the type 1 particle, after which the energy (and momentum) of the type 1 particle have changed.
- D: This shows diagrams B and C combined. The type 1 particle goes along all by itself, emits a type 2 particle, later on it reabsorbs the type 2 particle, and then goes along all by itself again.
That is the basic structure of how particles behave in physics. Energy (and momentum) conservation are always respected, so moving along each of the lines in the above diagrams each particle conserves its energy (and momentum), and at each of the vertices where a particle is emitted or absorbed the sum of the energies (or momenta) over all particles coming into the vertex is the same as the sum going out of the vertex.
Thus in diagram B the incoming type 1 particle's energy (and momentum) are shared between the outgoing type 1 and type 2 particles. This sharing between the type 1 and type 2 particles has to respect only the fact that the sum of the energies (or momenta) have to add up to the energy (or momentum) of the incoming type 1 particle. That means that one of the particles can have a negative energy and the other a positive energy, as long as the sum has the correct value.
The notion of a negative energy is counterintuitive. What does it mean? The physicist's definition of "energy" is the "frequency of oscillation" of the wave that is associated with the particle. Frequency can have any value (positive or negative) just as the rate of advance of a clock can be anything (forwards or backwards), so analogously energy can have any value.
Conservation of energy and momentum at the vertex where a particle is emitted or absorbed means that the particles don't have complete freedom to choose whatever energy and momentum they want to have, because their sum is constrained to be the same as whatever it was at the start. When a particle is going along all by itself (as in diagram A) there is a definite relationship between its energy and its momentum. After the particle has emitted another particle (as in diagram B), although the total energy and momentum are conserved the individual particles have energy and momentum that do not have the harmonious relationship that exists when the particle is going along all by itself. The physical consequence of this conflict in each particle is that they cannot not travel very far before they have to get their energy and momentum back into a harmonious relationship, and this requires the further emission or absorbtion of a second particle (as in diagram C). Diagram D brings it all together, where energy (and momentum) are conserved everywhere in the diagram (i.e. along each line, and passing through each vertex), and the conflict between each particle's energy and momentum in the "loop" part of the diagram means that the loop cannot be very large.
I think that this conflict between the energy and momentum of each particle, which is caused by the exact energy (and momentum) conservation everywhere in diagram D, is what is wrongly referred to (in popular journalese) as a temporary violation of energy (and momentum) conservation.
The relationship between the size of the conflict between each particle's energy and momentum and extent of the loop in diagram D is given by Heisenberg's uncertainty principle. The greater the conflict the smaller the loop. A particle going along all by itself has perfect consistency between its energy and momentum, so it is not part of a finite-sized loop.
posted by Stephen Luttrell @ 4:24 pm
17 Comments:
So is it expressly impossible to "steal" the energy from the loop part of the diagram ? Having injected an initial energy (that in diagram A)?
Your analogy with financial transactions is interesting, since I was originally motivated by arbitrage in financial markets to see if this kind of arbitrage can be done with energy in physical processes...
Energy is precisely conserved at each of the vertices in the diagram, so what comes in to each vertex is the same as what goes out of each vertex. In fact, we hardwire this property into our description of reality (i.e. physics) at a very low level, and this is then confirmed because it leads to lots of predictions that are consistent with experimental observation.
So arbitrage doesn't happen in physics, even though in popular journalese Heisenberg's uncertainty principle is usually described as if it does allow arbitrage to happen.
Your last remark about adding another vertex is essentially correct. Energy conservation means that the total energy has to be the same as before, which doesn't necessarily involve having negative energy states.
The only operations you can perform on particles are via vertices of the type that I have shown in the diagrams. Even Feynman's daemon is limited to operating this way! It is amazing that nature reduces to such a simple description, but all experimental observations indicate that this is indeed the case.
Nice post. It should also be noted that the energy/time Heissenberg uncertainty relationship is itself, kind of a finicky beast (what with there being no operator conjugate to the Hamiltonian.)
Energy/time and momentum/position mix with each other under Lorentz transformations, so they can't be considered independently. If you use scalars constructed out of 4-vectors (e.g. k.x) then it hides the distinction between the 0th component and the 1st-3rd components. I don't think that there is a problem specifically with energy/time uncertainty.
is there any violation of the uncertainity principle, suppose we use particles without momentum, which would give the position without altering the momentum of the target?
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A particle with zero momentum has infinite wavelength, so it cannot resolve any spatial detail at all, so it can't tell you the position of the target.
The intention of my original posting was to point out that the Heisenberg uncertainty principle is really a derived result, rather than a fundamental principle in its own right. That is why quoting HUP as your starting point often leads to wrong thinking and thence to wrong conclusions. It is surprising how many physicists erroneously think that energy and/or momentum conservation can be temporarily violated because of HUP.
The above makes sense. Might popular journalese, in speaking of “virtual particles,” recognize temporary violations of the arrow of time?
To the best of my recollection, popular journalese may be unclear about whether there is involved any “real” violation of the arrow of time.
Or, if in such case, and it is considered that there is a real violation of the arrow of time, I would be unclear about whether there may be some sort of superior, physical law of “conservation of reality” that allows minute particles (which otherwise, in “virtual respect,” would disobey the law of conservation of matter and energy) to temporarily violate the general arrow of time.
Must any apparent, relative violation of the arrow of time “not really” violate it?
Just for the record, the above comment arrived on 30 October 2007, approximately two years after the original posting.
I haven't mentioned "arrow of time" anywhere in my posting. In the case of elementary particle interactions there is no arrow of time, except for some minute effects due to violation of CP symmetry (which is equivalent to violation of T symmetry - see here).
Perhaps you are trying to make a connection with the comment you made on another of my postings Why time keeps going forwards. However, that posting I explained why one needs a large out-of-equilibrium system in order to exhibit directionality in time.
Elementary particle interactions do not in themselves exhibit time directionality, except for the CP-violating exception(s) noted above.
“The truth is that there is no violation of the law of energy conservation.
The lending or borrowing of energy (and momentum) is done in a way that always respects energy (and momentum) conservation.”
"Energy conservation means that the total energy has to be the same as before...."
Many thanks. It requires more work for an interested layman, such as myself, to appreciate many of these concepts. Being out of my depth, I, no doubt, am often prone to use words having precise meanings in a particular field contrary to the way they should be understood.
That said, I pause over the words “no violation,” “always,” and "before."
I take it there is, in time, never a failure for energy to be conserved.
So, what remains puzzling is how, in the case of elementary particle interactions, there is no arrow of time, yet there is ALWAYS conservation?
I take it there is some aspect of time involved, but that any direction for such time should be discounted for purposes of appreciating conservation of energy?
I take it there would be no measurable manifestation of loss of conservation, regardless of perspective taken for direction in time?
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BTW, I noticed someplace where you discussed spooky action at a distance.
I came across an interesting discussion about this at http://www.innerexplorations.com/catchmeta/mys1.htm.
There, it seems to be suggested that there is an “interior structure” to a subtle, ultimate ground of existence.
It was unclear whether empiricism should necessarily rule out such structure as being in respect of a sort of synchronizing Thing versus a Consciousness.
However, such concern may be inappropriate, outside of a forum that deals with unverifiable intuition.
You said:
I take it there is some aspect of time involved, but that any direction for such time should be discounted for purposes of appreciating conservation of energy?
That is correct! Conservation of energy (or more generally energy&momentum) follows from the invariance of the laws of physics under displacement in time (or more generally time&space). This is a particular instance of Noether's theorem.
So, if you shift your experimental apparatus in time&space then the results that you obtain are unchanged, assuming that your apparatus is isolated from external influences. This shift can be in any direction (forwards or backwards) in time&space, i.e. directionality is not hard-wired into the laws of physics.
As I said before, an exception to this is CP (or T) violation (see here).
So, it seems (?) conservation among conjugate pairs applies always and everywhere, to all changes in energy and momentum, in any direction of time or space.
But, if every change must be symmetrically balanced against an offset, must there not be an offset or mirror (?) universe or “conserved current” to our own ... and so on ....
If every change is required simultaneously to be balanced by an offset, then what is the offset in respect of which any change next in sequence is “caused” or instigated? (Perhaps, this may depend upon what the definition of “change” is.)
Must every change visited upon every world be imposed in discretely balanced units?
Must our universe, itself, as a whole, obey a unifying and balancing formula that limits expressions of degrees of symmetry and synchronicity, such that even our universe should be represented in respect of discrete, digitally quantifiable sequences of expression in time and space?
Should our entire universe of time, space, and/or time-space be considered as consisting of a sum series of discrete expressions of quanta, rather than as being continuous?
My thinking may seem only a variation on turtles all the way down. If so, no need to reply.
Yes, conservation of energy & momentum applies everywhere. Each elementary interaction that occurs must individually balance the books. There is no "mirror" universe to take up any "slack"; conservation is achieved entirely within our universe.
Conservation is a strong constraint on the types of interaction that are allowed to occur, because any possibilities that don't balance the books can be ruled out straight away. Also, conservation applies simultaneously and independently at every space-time point.
That answers most of your questions, apart from the one about discrete space-time. Currently, there is no experimental evidence that conflicts with our assumption that space-time is continuous.
“conservation applies simultaneously and independently at every space-time point”
I pause over the words “independently” and “point.”
As substitutes, I would have guessed “synchronously and interdependently” and “point and field.”
As I say, I have much to learn.
I may become less fuzzy as I search for sites to help me appreciate how particles may become polarized, entangled, and disentangled.
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MUSING ABOUT CAUSATION VS. CONSERVATION: On faith, though we perceive reality only indirectly, we take it that a conserved “physical reality” exists. After all, we do not sense even our bodies except in respect of sensations of it. As cause of such reality, the only asymmetrical, unbalanced source we seem able to make measurable sense out of is the Big Bang (considered at the beginning to have arisen from a point).
Every sensed event that sequences thereafter is conserved, in derivation or respect of the Big Bang. For empirical concerns, we seem better served to say that “it follows from A, that B,” etc., rather than to say “A causes B.”
Question: So, does B follow in complete, uninterrupted continuity, or does B follow in respect of discreteness among ultimate units or segments of space or time?
Well, that seems a question that cannot very well be tested empirically. If mortal time and space sequence only in discrete events, no mortal could very well test or observe anything “between” such events.
Unclear to my intuition is whether such a notion may be tested mathematically — such as if such a notion could further a unifying, mathematical model. For example, if B follows in respect of A, as a discrete segment, then Zeno’s paradox may be resolved.
But, such a resolution of Zeno would also seem to revive an alternative variation on “turtles all the way down.”
After all, what then would account for “causing” each next event after each next preceding event? (It seems recourse would have to be to “Goddidit,” since no testing could be done for what takes place “between” the segments.)
May one respect any world view without in some way respecting a “turtle” of some sort --- whether secular or sectarian?
A clever strategy may be to “be a perspective of the turtle.” After all, how can we do otherwise?
Were Dr. Seuss alive to write a story for prodigies of mathematics, he may have entitled it “Being the Turtle.”
All we can do is to try to explain how the universe appears to work, but we cannot explain why it works the way it does. All we can do is to observe things (whatever they are) with our senses (whatever they are), and then to improve our ability to predict the results of future observations.
Of course, it is an interesting question to ask what might be going on in places that we cannot observe, or to ask why things are the way they are. On this, all I can do is to quote Wittgenstein:
...and what we cannot talk about we must pass over in silence.
Question: If I have a slit experiment where the slit is a delta function the spread of momentum in a following measurement would be infinite. If I know the initial momentum of the particle, how is the conservation of energy/momentum held for any given measurement? The narrower the slit the more momentum is offered to the particle in the transverse direction.
Momentum is conserved because the object that contains the slit is also a part of the whole quantum mechanical system, i.e. object-with-slit + particle-going-through-slit. If the particle has a large transverse momentum, then the object-with-slit has a large transverse momentum in the opposite direction, and overall energy/momentum conservation is maintained.
The above experimental set-up is a macroscopic version of a particle being elastically scattered by a potential well.
You can represent all of these situations using Feynman diagrams to show the "particles" scattering off each other. The resulting interference pattern comes about because of the complex-valued sum over all of the intermediate states (e.g. all possible paths of the particle through the slits) that lead to each final state (e.g. the exact position of the particle hitting the screen, or whatever).
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