In recent 7 days, about 100 news outlets have informed their readers about "progress" in the construction of EM drive, and sometimes warp drive, that will allow NASA and everyone else to get everywhere (L.A., Moon, Mars, other galaxies) very quickly, perhaps faster than the speed of light.
Now the density of this nonsense has sufficiently decreased for me to find the courage to write a serious blog post: hundreds of flabbergasting imbeciles is what I am used to and ready to face.
All the hundreds of the news reports were derived from a post (about an "evaluation" of those things) on a website called NASAspaceFLIGHT.com which only "tries" to look credible, as you can easily find out. Why do hundreds of "science journalists" find it adequate to spread such big claims that only build on such a website?
There are two possible names of the cool technologies that will revolutionize spaceflight: "warp drive" and "EM drive". Warp drive is gravitational – it's supposed to curve the spacetime in such a way that the superluminal motion becomes possible – while the EM drive is basically electromagnetic (most science journalists don't care about and can't spot the difference between electromagnetism and gravity, anyway). Concerning the former, in July 2013, there was another wave of this warp drive nonsense in the media and I explained why it's completely wrong.
Special relativity bans faster-than-light motion of massive bodies because it says that in all other inertial frames, physical phenomena may be described by the same and equally simple laws of physics. But if you transform a superluminal motion to a different inertial system by the Lorentz transformation, it turns into the motion that goes backwards in time! That's a problem because the existence of the superluminal spaceships would imply the existence of spaceships flying backwards in time (those can be mapped to each other by Lorentz transformations) and it would allow you to castrate your grandfather before he sleeps with your grandmother, and that would make (or have made or would have been made or whatever) your existence impossible and our Universe logically inconsistent.
Some people want to believe that general relativity allows one to "circumvent" the limitation on speed that results from special relativity. But this ain't the case. Special relativity is still firmly incorporated in general relativity and shows its muscles in many ways.
First, special relativity holds locally. Small regions of spacetime (much smaller than the curvature radius length scale) look like the flat Minkowski space and special relativity holds there. Warp drive is supposed to curve the spacetime so that it makes the space in front of you "shorter" and easier to fly through, while the irrelevant space behind you is "longer". Warp drive apologists think that it's possible – general relativity may curve the space in any way – but they're wrong.
To get this kind of curvature, you need a negative energy density at least somewhere. You need to violate some energy conditions. I wrote a text about energy conditions and warp drive on Quora which you may read to see some details.
The negative energy density is ultimately forbidden because if it were allowed, the vacuum would be unstable. Equivalently, it could create particles (tachyons) that move faster than light and revive the aforementioned grandfather paradoxes.
There is another way to see that spaceships can't fly faster than light. From the viewpoint of very long distance scales, the space looks flat and empty – up to small and local perturbations (objects etc.). So it's a Minkowski space and the behavior of the local perturbations (objects etc.) must obey the Lorentz invariance inherited from the surrounding space, too. For this reason, it doesn't matter at all whether the local perturbations (objects such as spaceships) involve some local curvature of the spacetime (as a part of their design) or not. They're still some local objects or perturbations. Their motion that would surpass the speed of light is forbidden by the rules of special relativity because those still apply in the almost empty surrounding space!
If some strong curvature were enough to defeat special relativity, special relativity would be completely wrong. For example, black holes or some clever bound states of black holes could move faster than light. But if black holes could do it, all elementary particles could also do it – to one extent or another – because elementary particles may be viewed as "very tiny black holes" (lighter than the Planck mass) for which the quantum corrections become very important.
So the existence of curvature doesn't really change anything whatever about the fact that the superluminal motion is prohibited in Nature. Some people just don't like (special) relativity because it "restricts" them and that's bad (the fact that the laws of Nature always restrict you and everything else must be eluding them). So they invent an ideology based on a wishful thinking or belief that the "next" theory must surely eliminate the rules that special relativity has brought us – in this case that the speed can't exceed the speed of light.
But that's not how science works. Science doesn't go through similar counterrevolutions. The advance known as "special relativity" really meant that "non-relativistic physics" with its simple-minded possibilities (including arbitrarily high speeds) has been falsified. Falsification is really an irreversible process. General relativity doesn't mean and couldn't mean that physics would return to the state when all speeds are allowed once again. If that were so, it would really mean that general relativity restores Newtonian physics – but the latter had been killed since 1905.
Reactionless drive
The case of EM drive is analogous. What is EM drive? The proponents and fans talk about some microwaves in a cavity that push you without any propellant. It's clearly an example of a reactionless drive. A gadget sits in the middle of the empty space. Someone pushes a button and it suddenly starts to accelerate. No, that's impossible because it violates the momentum conservation law or the third Newton's law, if you wish.
Some people will tell you that the law isn't violated because "the vacuum" is what gives the momentum to the spaceship. But that's nonsense. By definition, the vacuum doesn't carry any momentum – its momentum is zero both in the initial state and the final state because it's really the same state, the vacuum state. If it looks like the object is accelerating itself and nothing goes out of it (and if it quacks in the same way etc.), it's because it is accelerating itself and nothing goes out of it. This is a straight denial of the momentum conservation law, and that's why this spaceship is forbidden.
Now, the individuals behind this particular "breakthrough" say that they combine some random ingredients – a church bell, microwaves, a superconductor, a cherry pie, the U.S. constitution, a witch's gray hair, and a few others – and they do a certain measurement and obtain a nonzero force in micronewtons, or something like that, which they can't explain. (Even if it worked in principle, such forces could be attributed to a mosquito and be enough to move a mosquito but all of these people already speak about getting to Mars in seconds.)
Well, that shouldn't be surprising that these people can't explain the force. People doing these things are complete idiots so biology predicts that they can't calculate the force they should measure in a complex experiment and they can't explain anything in physics or engineering. So the observations exactly agree with the predictions by physics and biology: science predicts that these people will behave like complete idiots because they're complete idiots, and that's exactly what is observed, indeed, because these people behave as complete idiots, too.
Lots of fans of this nonsense will tell you that quantum electrodynamics and its "mysterious vacuum" will allow you to violate the energy conservation law. This wishful thinking (or, more precisely, the wishful absence of thinking) is completely analogous to the absence of thinking of the warp drive fans who think that general relativity "abolishes" the special relativistic limitation on speed.
There is absolutely no reason to think that quantum mechanics or quantum electrodynamics or their effects should "abolish" the momentum conservation law or to provide you with some exceptions. As we learned from Emmy Noether exactly 100 years ago, the momentum conservation law is just the other face of the invariance of the laws of physics under spatial translations. The laws of physics are the same here and on another place. And that's enough to see that a conserved quantity – which we call momentum – has to exist.
None of these relationships is modified by quantum mechanics. We have \([H,\vec p]=0\) which may be interpreted in two different ways. Either the momentum's time derivative is zero, thanks to the Heisenberg equations, so the momentum doesn't change in time (at all: it is an operator equation). Or the Hamiltonian \(H\) isn't changed if you transform it by the transformations generated by \(\vec p\), namely by spatial translations. Because \(H\) defines the dynamical laws of physics, the previous sentence is equivalent to saying that the laws of physics are translationally symmetric.
So many people are behaving completely irrationally. There is absolutely no reason why the "next layer" of a physical theory should "abolish" the particular insights of previous theories such as the momentum conservation law. But some people want to believe that any mystery that is incomprehensible to their eyes and peabrains – and needless to say, none of these people understands quantum mechanics – will make all their dreams come true.
An Islamist doesn't know what QCD is but if you vaguely tell him what QCD is, he will surely believe that QCD is able to calculate that he will get exactly 72.0 virgins in the heaven after he performs a terrorist attack. Well, that's not what QCD does. There is absolutely no reason why it should. Very analogously, there is absolutely no reason why general relativity should "reallow" superluminal motion or why quantum mechanics should "allow" violations of the momentum conservation law.
In any quantum mechanical theory, if a conserved quantity \(L\) (either \(H\) or another observable that commutes with \(H\) – in our case, \(\vec p\)) is known for certain to have the value \(V\) after a measurement, it is guaranteed that it will have exactly the same value \(V\) up to the next measurement. It's because the wave function in between these two measurements is an eigenstate of \(L\) with the eigenvalue \(V\):\[
L \ket\psi = V \ket \psi
\] When you make the second measurement, it depends what you measure. If you measure \(L\) or another observable that commutes with \(L\), \(\ket\psi\) will remain an eigenstate of \(L\) with the same eigenvalue \(V\). If you measure a different observable \(M\) that doesn't commute with \(L\), such a measurement inevitably influences the value of \(L\) – you can't measure things without distorting the measured object. In that case, the value of \(L\) no longer has to be \(V\) after the second measurement.
But if you talk about the total momentum of the observed system and the observer, their total momentum is exactly conserved – it may be verified by an external observer. There is no way to circumvent this fact. This fact holds in any quantum mechanical theory where \([L,H]=0\). And quantum electrodynamics, QED, is just another sophisticated example of such a theory. No, the energy conservation law cannot be violated.
People love to be extremely sloppy about the "character of the uncertainties" that exist in quantum mechanics. They think that the uncertainty principle surely allows them to violate any conservation law by a little bit. But it doesn't. What the uncertainty principle says is that by measuring \(M\) such that \([L,M]\neq 0\), you affect \(L\), so you no longer know \(L\) after the measurement.
But if you know \(L\) and if \([L,H]=0\), then you continue to know \(L\) up to the next (final?) measurement! In the case of the conserved energy (imagine the energy of the harmonic oscillator), the energy may be written as a function of positions \(x\) and momenta \(p\). In classical physics, you could measure all components of \(x\) and \(p\) to calculate \(H\).
In quantum mechanics, \(H\) can't be measured by measuring \(x\) and \(p\) because \(x\) and \(p\) can't be measured at the same moment. For macroscopic objects, the uncertainty principle only inserts a tiny uncertainty about the values of \(x\) and \(p\) – and their functions such as \(H\). But in the microscopic case and if you want to be completely accurate, it's important to realize that each observable – such as \(H\) – must be measured by a very special, separate procedure, and you can never imagine that the measurement of all conceivable observables may be reduced to the "universal" measurement of \(x\) and \(p\) and some \(H\)-dependent calculation. That's how classical physics worked but quantum mechanics doesn't work in this way! Every observable – e.g. every function of the operators \(x\) and \(p\) – is measured in a fundamentally different way.
Casimir effect and superstitions
On Monday, it's been 15 years since the death of Hendrik Casimir whom we remember for the Casimir invariants and the Casimir effect. I wanted to write a biography on Monday but I was still overwhelmed by the amount of EM drive or warp drive nonsense and wanted to avoid the topic while the concentration and self-confidence of crackpots was still high.
The Casimir effect is the purely quantum mechanical attraction between two parallel metallic plates whose distance is \(a\). The electric potential obeys \(\phi=0\) at both surfaces. This implies that in between the plates, in the interval \(0\leq z \leq a\), the electromagnetic waves may be expanded into discrete Fourier series instead of the continuous Fourier transform.
This "modification of the vacuum" changes the counting of the zero-point energy of the quantum harmonic oscillators carried by the electromagnetic field. With the two metallic plates, the vacuum energy density per unit area of the plates includes a sum over the Fourier series integer \(n\)\[
\eq{
\frac{E_0}{A} &= -K \sum_{n=1}^\infty \zav{ \frac{n}{a} }^3 =\\
&= -\frac{K}{a^3} (1^3+2^3+3^3+\dots) = -\frac{K}{120 a^3}
}
\] I have used the fact that the sum of third powers of positive integers is \(\zeta(-3)=+1/120\). It is a completely analogous result to the fact that the sum of positive integers equals \(-1/12\).
If you care, the full derivation gives you the normalization constant \(K=\hbar c \pi^2/6\).
So the energy is negative and extremely important for \(a\to 0\), and that's why the two plates attract by a force that scales like \(1/a^4\). This is the Casimir force.
The displayed formula above showed that the "energy density has a negative piece" which was attributed to the modified zero-point energies. The deluded fans of various EM drive nonsense (which is a group that, under certain circumstances, may include people like Kip Thorne and Stephen Hawking as well) totally misrepresent what this "negative piece" in energy density means, can mean, or cannot mean.
They claim that it is a step towards allowing negative energy densities in macroscopic situations, too.
But once again, this claim is completely wrong. The nonzero energy density in between the plates only arose due to quantum mechanics – it was proportional to \(\hbar\) i.e. to Planck's constant. So it is guaranteed to go to zero in the classical limit, e.g. if the distances between the plates become macroscopic.
Note that we were really not calculating the interaction energy of the two plates as a volume integral. Instead, we calculated the "total" value of the integrated energy for the vacuum state. So we can't really say what places contributed the negative piece that produced the Casimir effect. The conceptually correct answer to this question is that it is the metallic plates themselves that are necessary for the modification of the zero-point energies, and in a natural accounting, the plates themselves – with their modified boundary conditions for the electromagnetic fields – contribute the negative Casimir energy.
How we attribute the energy to regions is a question that slightly depends on conventions. Note that in Maxwell's theory, there are questions what the "right" stress-energy tensor should be. There are several options. Only the total (integrated) energy and momentum are conserved and convention-independent.
But whatever convention you choose, you shouldn't forget about the fact that the very small distance between the two metallic plates was a necessary condition for producing a sizable negative contribution to the ground state energy. If you make the cavities macroscopic, the effect goes to zero. And even when and if it is not zero, the force is only a new contribution to the force between two plates (just like the van der Waals force which is a related example of a quantum-mechanics-induced force). And the force between two plates obeys the third Newton's law, of course. The action and the reaction match.
Let me describe an analogy with quantum tunneling. In quantum tunneling, a particle is able to penetrate walls that would be classically impenetrable. It means that classically, the particle would need a negative kinetic energy while being in the wall, but the kinetic energy is \(p^2/2m\) which is a positively definite operator. Classically, the particle just can't be "in the wall" and you could be tempted to say that this is true even quantum mechanically because \(p^2/2m\) is a positively definite operator.
However, the uncertainty principle offers you a loophole here. If you ask whether the particle may be "in the wall", you know its position well enough, so there's some unavoidable error in the momentum. You may imagine the momentum to be small and imaginary. And that's enough for the particle to get through if it is quick enough. The appearance of the imaginary momentum may be controversial but I just used these words to describe the exponential decrease of the wave function inside the barrier.
However, the exponentially growing/decreasing wave function can't be extrapolated to the whole space because it would fail to be normalizable (it wouldn't even be normalizable to the delta-function). Equivalently, the momentum is a Hermitian operator so all of its eigenvalues (and expectation values) are real. That's why the description in terms of the "imaginary momentum" only has a limited validity, "inside the barrier", and one must be careful what we mean.
The case of the Casimir effect may be described in an analogous way. Locally, we may imagine that the electromagnetic field has \(\bra 0 E^2 \ket 0\lt 0\) or \(\bra 0 B^2\ket 0\lt 0\) at some points in between the plates. However, this description in terms of "imaginary electromagnetic fields" can't be extrapolated to macroscopic regions of space. Such things only occur due to the small distances between the metals.
At big regions of the space, you must be able to write the electromagnetic fields as the Fourier transforms allowing all real frequencies and the energy carried by those are positively definite again.
All of this may be confusing and I can't prove a rigorous "no-go theorem" for similar gadgets because I don't really know what the "most general gadget of this kind" that is supposed to exploit the "negativity of the Casimir energy" is supposed to do. But what I can do is to prove that completely mundane and ordinary arrangements of matter that are being shown as EM drive don't display any miraculous behavior. They're so big and ordinary that classical physics is really enough, classical physics works, and it prohibits the self-acceleration and similar wonders.
I have watched several minutes of talks by Harold Sonny White and one more guy who "invented" EM drive. It seems fair to compare them to the pseudoscience supported by Rudolph II when he was the Austrian Emperor with headquarters in Prague. I think that the guys who worked for Rudolph II – inventing elixir of youth and transmutation of elements to gold and lots of similar wonderful stuff – were actually much more scientific than the men behind EM drive and (Albucierre) warp drive.
They had at least some stories, some theories that have apparently passed some tests, some excuses why the latest elixir didn't work or why no gold emerged in their test tube. The theories were ultimately wrong and people could have seen that they were wrong but they have at least tried.
The men behind EM drive and warp drive don't have anything of the sort. They just randomly combine building blocks and hope that this will lead to some breakthrough. Once you get used to the nonsensical combinations of physics buzzwords that don't fit together at all, their talks are boring and virtually content-free – and some of their fans interpret this lack of content as their being careful researchers. Sorry, they are just vacuous inarticulate crackpots. They may have been employed by NASA but they're still pathetic morons, much like all of their apologists.
Off-topic, legit technology: GPS with sub-inch accuracy (100 times better) will come to your smartphone soonAlmost all the writers indicate that the information is trustworthy. Exceptions are rare. Ethan Siegel at Forbes is an example. The Skeptics Guide. Katie Palmer of Wired must also be praised for her explanations why it's still poppycock. (Needless to say, hundreds of aggressive kooks attacked Palmer (pic) and upvoted themselves in the comments. By upvotes, kooks defeated sane readers 100-to-1. Be aware that poppycock isn't a cock that pops, as they think, but an original Dutch word for a soft i.e. near-liquid Å¡it.) Numerous comments in the Tesla battery thread were dedicated to this warp drive hype. I was overwhelmed and couldn't find energy to write a separate blog post about that topic. I didn't want to be attacked by thousands of flabbergasting imbeciles.
Now the density of this nonsense has sufficiently decreased for me to find the courage to write a serious blog post: hundreds of flabbergasting imbeciles is what I am used to and ready to face.
All the hundreds of the news reports were derived from a post (about an "evaluation" of those things) on a website called NASAspaceFLIGHT.com which only "tries" to look credible, as you can easily find out. Why do hundreds of "science journalists" find it adequate to spread such big claims that only build on such a website?
There are two possible names of the cool technologies that will revolutionize spaceflight: "warp drive" and "EM drive". Warp drive is gravitational – it's supposed to curve the spacetime in such a way that the superluminal motion becomes possible – while the EM drive is basically electromagnetic (most science journalists don't care about and can't spot the difference between electromagnetism and gravity, anyway). Concerning the former, in July 2013, there was another wave of this warp drive nonsense in the media and I explained why it's completely wrong.
Special relativity bans faster-than-light motion of massive bodies because it says that in all other inertial frames, physical phenomena may be described by the same and equally simple laws of physics. But if you transform a superluminal motion to a different inertial system by the Lorentz transformation, it turns into the motion that goes backwards in time! That's a problem because the existence of the superluminal spaceships would imply the existence of spaceships flying backwards in time (those can be mapped to each other by Lorentz transformations) and it would allow you to castrate your grandfather before he sleeps with your grandmother, and that would make (or have made or would have been made or whatever) your existence impossible and our Universe logically inconsistent.
Some people want to believe that general relativity allows one to "circumvent" the limitation on speed that results from special relativity. But this ain't the case. Special relativity is still firmly incorporated in general relativity and shows its muscles in many ways.
First, special relativity holds locally. Small regions of spacetime (much smaller than the curvature radius length scale) look like the flat Minkowski space and special relativity holds there. Warp drive is supposed to curve the spacetime so that it makes the space in front of you "shorter" and easier to fly through, while the irrelevant space behind you is "longer". Warp drive apologists think that it's possible – general relativity may curve the space in any way – but they're wrong.
To get this kind of curvature, you need a negative energy density at least somewhere. You need to violate some energy conditions. I wrote a text about energy conditions and warp drive on Quora which you may read to see some details.
The negative energy density is ultimately forbidden because if it were allowed, the vacuum would be unstable. Equivalently, it could create particles (tachyons) that move faster than light and revive the aforementioned grandfather paradoxes.
There is another way to see that spaceships can't fly faster than light. From the viewpoint of very long distance scales, the space looks flat and empty – up to small and local perturbations (objects etc.). So it's a Minkowski space and the behavior of the local perturbations (objects etc.) must obey the Lorentz invariance inherited from the surrounding space, too. For this reason, it doesn't matter at all whether the local perturbations (objects such as spaceships) involve some local curvature of the spacetime (as a part of their design) or not. They're still some local objects or perturbations. Their motion that would surpass the speed of light is forbidden by the rules of special relativity because those still apply in the almost empty surrounding space!
If some strong curvature were enough to defeat special relativity, special relativity would be completely wrong. For example, black holes or some clever bound states of black holes could move faster than light. But if black holes could do it, all elementary particles could also do it – to one extent or another – because elementary particles may be viewed as "very tiny black holes" (lighter than the Planck mass) for which the quantum corrections become very important.
So the existence of curvature doesn't really change anything whatever about the fact that the superluminal motion is prohibited in Nature. Some people just don't like (special) relativity because it "restricts" them and that's bad (the fact that the laws of Nature always restrict you and everything else must be eluding them). So they invent an ideology based on a wishful thinking or belief that the "next" theory must surely eliminate the rules that special relativity has brought us – in this case that the speed can't exceed the speed of light.
But that's not how science works. Science doesn't go through similar counterrevolutions. The advance known as "special relativity" really meant that "non-relativistic physics" with its simple-minded possibilities (including arbitrarily high speeds) has been falsified. Falsification is really an irreversible process. General relativity doesn't mean and couldn't mean that physics would return to the state when all speeds are allowed once again. If that were so, it would really mean that general relativity restores Newtonian physics – but the latter had been killed since 1905.
Reactionless drive
The case of EM drive is analogous. What is EM drive? The proponents and fans talk about some microwaves in a cavity that push you without any propellant. It's clearly an example of a reactionless drive. A gadget sits in the middle of the empty space. Someone pushes a button and it suddenly starts to accelerate. No, that's impossible because it violates the momentum conservation law or the third Newton's law, if you wish.
Some people will tell you that the law isn't violated because "the vacuum" is what gives the momentum to the spaceship. But that's nonsense. By definition, the vacuum doesn't carry any momentum – its momentum is zero both in the initial state and the final state because it's really the same state, the vacuum state. If it looks like the object is accelerating itself and nothing goes out of it (and if it quacks in the same way etc.), it's because it is accelerating itself and nothing goes out of it. This is a straight denial of the momentum conservation law, and that's why this spaceship is forbidden.
Now, the individuals behind this particular "breakthrough" say that they combine some random ingredients – a church bell, microwaves, a superconductor, a cherry pie, the U.S. constitution, a witch's gray hair, and a few others – and they do a certain measurement and obtain a nonzero force in micronewtons, or something like that, which they can't explain. (Even if it worked in principle, such forces could be attributed to a mosquito and be enough to move a mosquito but all of these people already speak about getting to Mars in seconds.)
Well, that shouldn't be surprising that these people can't explain the force. People doing these things are complete idiots so biology predicts that they can't calculate the force they should measure in a complex experiment and they can't explain anything in physics or engineering. So the observations exactly agree with the predictions by physics and biology: science predicts that these people will behave like complete idiots because they're complete idiots, and that's exactly what is observed, indeed, because these people behave as complete idiots, too.
Lots of fans of this nonsense will tell you that quantum electrodynamics and its "mysterious vacuum" will allow you to violate the energy conservation law. This wishful thinking (or, more precisely, the wishful absence of thinking) is completely analogous to the absence of thinking of the warp drive fans who think that general relativity "abolishes" the special relativistic limitation on speed.
There is absolutely no reason to think that quantum mechanics or quantum electrodynamics or their effects should "abolish" the momentum conservation law or to provide you with some exceptions. As we learned from Emmy Noether exactly 100 years ago, the momentum conservation law is just the other face of the invariance of the laws of physics under spatial translations. The laws of physics are the same here and on another place. And that's enough to see that a conserved quantity – which we call momentum – has to exist.
None of these relationships is modified by quantum mechanics. We have \([H,\vec p]=0\) which may be interpreted in two different ways. Either the momentum's time derivative is zero, thanks to the Heisenberg equations, so the momentum doesn't change in time (at all: it is an operator equation). Or the Hamiltonian \(H\) isn't changed if you transform it by the transformations generated by \(\vec p\), namely by spatial translations. Because \(H\) defines the dynamical laws of physics, the previous sentence is equivalent to saying that the laws of physics are translationally symmetric.
So many people are behaving completely irrationally. There is absolutely no reason why the "next layer" of a physical theory should "abolish" the particular insights of previous theories such as the momentum conservation law. But some people want to believe that any mystery that is incomprehensible to their eyes and peabrains – and needless to say, none of these people understands quantum mechanics – will make all their dreams come true.
An Islamist doesn't know what QCD is but if you vaguely tell him what QCD is, he will surely believe that QCD is able to calculate that he will get exactly 72.0 virgins in the heaven after he performs a terrorist attack. Well, that's not what QCD does. There is absolutely no reason why it should. Very analogously, there is absolutely no reason why general relativity should "reallow" superluminal motion or why quantum mechanics should "allow" violations of the momentum conservation law.
In any quantum mechanical theory, if a conserved quantity \(L\) (either \(H\) or another observable that commutes with \(H\) – in our case, \(\vec p\)) is known for certain to have the value \(V\) after a measurement, it is guaranteed that it will have exactly the same value \(V\) up to the next measurement. It's because the wave function in between these two measurements is an eigenstate of \(L\) with the eigenvalue \(V\):\[
L \ket\psi = V \ket \psi
\] When you make the second measurement, it depends what you measure. If you measure \(L\) or another observable that commutes with \(L\), \(\ket\psi\) will remain an eigenstate of \(L\) with the same eigenvalue \(V\). If you measure a different observable \(M\) that doesn't commute with \(L\), such a measurement inevitably influences the value of \(L\) – you can't measure things without distorting the measured object. In that case, the value of \(L\) no longer has to be \(V\) after the second measurement.
But if you talk about the total momentum of the observed system and the observer, their total momentum is exactly conserved – it may be verified by an external observer. There is no way to circumvent this fact. This fact holds in any quantum mechanical theory where \([L,H]=0\). And quantum electrodynamics, QED, is just another sophisticated example of such a theory. No, the energy conservation law cannot be violated.
People love to be extremely sloppy about the "character of the uncertainties" that exist in quantum mechanics. They think that the uncertainty principle surely allows them to violate any conservation law by a little bit. But it doesn't. What the uncertainty principle says is that by measuring \(M\) such that \([L,M]\neq 0\), you affect \(L\), so you no longer know \(L\) after the measurement.
But if you know \(L\) and if \([L,H]=0\), then you continue to know \(L\) up to the next (final?) measurement! In the case of the conserved energy (imagine the energy of the harmonic oscillator), the energy may be written as a function of positions \(x\) and momenta \(p\). In classical physics, you could measure all components of \(x\) and \(p\) to calculate \(H\).
In quantum mechanics, \(H\) can't be measured by measuring \(x\) and \(p\) because \(x\) and \(p\) can't be measured at the same moment. For macroscopic objects, the uncertainty principle only inserts a tiny uncertainty about the values of \(x\) and \(p\) – and their functions such as \(H\). But in the microscopic case and if you want to be completely accurate, it's important to realize that each observable – such as \(H\) – must be measured by a very special, separate procedure, and you can never imagine that the measurement of all conceivable observables may be reduced to the "universal" measurement of \(x\) and \(p\) and some \(H\)-dependent calculation. That's how classical physics worked but quantum mechanics doesn't work in this way! Every observable – e.g. every function of the operators \(x\) and \(p\) – is measured in a fundamentally different way.
Casimir effect and superstitions
On Monday, it's been 15 years since the death of Hendrik Casimir whom we remember for the Casimir invariants and the Casimir effect. I wanted to write a biography on Monday but I was still overwhelmed by the amount of EM drive or warp drive nonsense and wanted to avoid the topic while the concentration and self-confidence of crackpots was still high.
The Casimir effect is the purely quantum mechanical attraction between two parallel metallic plates whose distance is \(a\). The electric potential obeys \(\phi=0\) at both surfaces. This implies that in between the plates, in the interval \(0\leq z \leq a\), the electromagnetic waves may be expanded into discrete Fourier series instead of the continuous Fourier transform.
This "modification of the vacuum" changes the counting of the zero-point energy of the quantum harmonic oscillators carried by the electromagnetic field. With the two metallic plates, the vacuum energy density per unit area of the plates includes a sum over the Fourier series integer \(n\)\[
\eq{
\frac{E_0}{A} &= -K \sum_{n=1}^\infty \zav{ \frac{n}{a} }^3 =\\
&= -\frac{K}{a^3} (1^3+2^3+3^3+\dots) = -\frac{K}{120 a^3}
}
\] I have used the fact that the sum of third powers of positive integers is \(\zeta(-3)=+1/120\). It is a completely analogous result to the fact that the sum of positive integers equals \(-1/12\).
If you care, the full derivation gives you the normalization constant \(K=\hbar c \pi^2/6\).
So the energy is negative and extremely important for \(a\to 0\), and that's why the two plates attract by a force that scales like \(1/a^4\). This is the Casimir force.
The displayed formula above showed that the "energy density has a negative piece" which was attributed to the modified zero-point energies. The deluded fans of various EM drive nonsense (which is a group that, under certain circumstances, may include people like Kip Thorne and Stephen Hawking as well) totally misrepresent what this "negative piece" in energy density means, can mean, or cannot mean.
They claim that it is a step towards allowing negative energy densities in macroscopic situations, too.
But once again, this claim is completely wrong. The nonzero energy density in between the plates only arose due to quantum mechanics – it was proportional to \(\hbar\) i.e. to Planck's constant. So it is guaranteed to go to zero in the classical limit, e.g. if the distances between the plates become macroscopic.
Note that we were really not calculating the interaction energy of the two plates as a volume integral. Instead, we calculated the "total" value of the integrated energy for the vacuum state. So we can't really say what places contributed the negative piece that produced the Casimir effect. The conceptually correct answer to this question is that it is the metallic plates themselves that are necessary for the modification of the zero-point energies, and in a natural accounting, the plates themselves – with their modified boundary conditions for the electromagnetic fields – contribute the negative Casimir energy.
How we attribute the energy to regions is a question that slightly depends on conventions. Note that in Maxwell's theory, there are questions what the "right" stress-energy tensor should be. There are several options. Only the total (integrated) energy and momentum are conserved and convention-independent.
But whatever convention you choose, you shouldn't forget about the fact that the very small distance between the two metallic plates was a necessary condition for producing a sizable negative contribution to the ground state energy. If you make the cavities macroscopic, the effect goes to zero. And even when and if it is not zero, the force is only a new contribution to the force between two plates (just like the van der Waals force which is a related example of a quantum-mechanics-induced force). And the force between two plates obeys the third Newton's law, of course. The action and the reaction match.
Let me describe an analogy with quantum tunneling. In quantum tunneling, a particle is able to penetrate walls that would be classically impenetrable. It means that classically, the particle would need a negative kinetic energy while being in the wall, but the kinetic energy is \(p^2/2m\) which is a positively definite operator. Classically, the particle just can't be "in the wall" and you could be tempted to say that this is true even quantum mechanically because \(p^2/2m\) is a positively definite operator.
However, the uncertainty principle offers you a loophole here. If you ask whether the particle may be "in the wall", you know its position well enough, so there's some unavoidable error in the momentum. You may imagine the momentum to be small and imaginary. And that's enough for the particle to get through if it is quick enough. The appearance of the imaginary momentum may be controversial but I just used these words to describe the exponential decrease of the wave function inside the barrier.
However, the exponentially growing/decreasing wave function can't be extrapolated to the whole space because it would fail to be normalizable (it wouldn't even be normalizable to the delta-function). Equivalently, the momentum is a Hermitian operator so all of its eigenvalues (and expectation values) are real. That's why the description in terms of the "imaginary momentum" only has a limited validity, "inside the barrier", and one must be careful what we mean.
The case of the Casimir effect may be described in an analogous way. Locally, we may imagine that the electromagnetic field has \(\bra 0 E^2 \ket 0\lt 0\) or \(\bra 0 B^2\ket 0\lt 0\) at some points in between the plates. However, this description in terms of "imaginary electromagnetic fields" can't be extrapolated to macroscopic regions of space. Such things only occur due to the small distances between the metals.
At big regions of the space, you must be able to write the electromagnetic fields as the Fourier transforms allowing all real frequencies and the energy carried by those are positively definite again.
All of this may be confusing and I can't prove a rigorous "no-go theorem" for similar gadgets because I don't really know what the "most general gadget of this kind" that is supposed to exploit the "negativity of the Casimir energy" is supposed to do. But what I can do is to prove that completely mundane and ordinary arrangements of matter that are being shown as EM drive don't display any miraculous behavior. They're so big and ordinary that classical physics is really enough, classical physics works, and it prohibits the self-acceleration and similar wonders.
I have watched several minutes of talks by Harold Sonny White and one more guy who "invented" EM drive. It seems fair to compare them to the pseudoscience supported by Rudolph II when he was the Austrian Emperor with headquarters in Prague. I think that the guys who worked for Rudolph II – inventing elixir of youth and transmutation of elements to gold and lots of similar wonderful stuff – were actually much more scientific than the men behind EM drive and (Albucierre) warp drive.
They had at least some stories, some theories that have apparently passed some tests, some excuses why the latest elixir didn't work or why no gold emerged in their test tube. The theories were ultimately wrong and people could have seen that they were wrong but they have at least tried.
The men behind EM drive and warp drive don't have anything of the sort. They just randomly combine building blocks and hope that this will lead to some breakthrough. Once you get used to the nonsensical combinations of physics buzzwords that don't fit together at all, their talks are boring and virtually content-free – and some of their fans interpret this lack of content as their being careful researchers. Sorry, they are just vacuous inarticulate crackpots. They may have been employed by NASA but they're still pathetic morons, much like all of their apologists.
EM drive, warp drive, gullibility without limits
![EM drive, warp drive, gullibility without limits]()
Reviewed by DAL
on
May 06, 2015
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