The classical aether vs. the modern vacuum

[genneth]

My problem is that I'd prefer to deepen my knowledge of standard QFT before I delve into non-standard works with about 400 pages...

So genneth told me that it is well accepted that the states we primarily describe in theory are not the ones we measure. What key words do I have to search for or, more specifically, which chapter do I have to read in one of the standard textbooks in order to understand this better ?

Useful keywords might be something along the lines of "locality of measurement in QFT". The basic problem is that operators which measure the Fock states need to have infinite spacetime extent. This paper explains it beautifully: http://arxiv.org/abs/gr-qc/0409054 Unfortunately it's not published, but it does reference papers that talk more about the problem (and less about the solution) which are published.

For more stuff on criticism of QFT and quantum mechanics in general: http://arxiv.org/abs/quant-ph/0609163

It's important to read these things critically however: authors often have a bias even when they don't want to. That one above has a heavy dose of Bohm.

 

[OOO]

Useful keywords might be something along the lines of "locality of measurement in QFT". The basic problem is that operators which measure the Fock states need to have infinite spacetime extent. This paper explains it beautifully: http://arxiv.org/abs/gr-qc/0409054 Unfortunately it's not published, but it does reference papers that talk more about the problem (and less about the solution) which are published.

For more stuff on criticism of QFT and quantum mechanics in general: http://arxiv.org/abs/quant-ph/0609163

It's important to read these things critically however: authors often have a bias even when they don't want to. That one above has a heavy dose of Bohm.

Thanks, I actually had textbooks in mind when I remembered you telling about things being "well accepted". But I'll take a look at those. Yes, I'm aware of that Bohm bias. I have also dedicated some time to studying Bohm and although interesting one has to take it or rather its advocates with a grain of salt.

 

[genneth]

Thanks, I actually had textbooks in mind when I remembered you telling about things being "well accepted". But I'll take a look at those. Yes, I'm aware of that Bohm bias. I have also dedicated some time to studying Bohm and although interesting one has to take it or rather its advocates with a grain of salt.

Yes, I would have preferred to give a textbook reference too. Unfortunately, textbooks tend to not deal with issues which are currently accepted as issues, but for which an accepted resolution is not available -- after all, it's more confusing for someone trying to learn from it. It's one of those things where almost everyone you talk to knows about the problem, but no one seems to know where is an authoritative source. I'm going to keep hunting for a while, to see if I can find something...

 

[meopemuk]

But isn't the time evolution defined by adding up all "path" amplitudes ?

The most fundamental way to define time evolution in quantum mechanics or QFT is through the Hamiltonian. It is not an accident that the Hamiltonian is also called "the generator of time translations". If $| \Psi(0) \rangle$ is a state vector at time t=0, then at later time the state vector is (in the Schroedinger representation)

$$|\Psi(t) \rangle = \exp(\frac{i}{\hbar}Ht) |\Psi(0) \rangle$$...(1)

Perhaps you can describe the time evolution in an alternative way (with path integrals, etc.), but whatever you do it should be equivalent to eq. (1).

Now, the bare particle Hamiltonian of QFT will immediately lead to a trouble with eq. (1). For example, in the creation/annihilation operator representation the Hamiltonian of QED has the form

$$H = H_0 + a^{\dag}b^{\dag}c^{\dag} + \ldots$$...(2)

where $H_0$ is the free particle Hamiltonian, $a^{\dag}b^{\dag}c^{\dag}$ describes simultaneous creation of 3 particles (electron + positron + photon), and the ellipsis denotes (a lot of) other terms, which are not relevant at this point. Substituting (2) into (1) we can obtain the following time evolution of the bare vacuum state

$$|0(t) \rangle = \exp(\frac{i}{\hbar}Ht) |0 \rangle \approx (1+ \frac{it}{\hbar}(H_0 + a^{\dag}b^{\dag}c^{\dag}) |0 \rangle = |0 \rangle + \frac{it}{\hbar} a^{\dag}b^{\dag}c^{\dag} |0 \rangle$$

So, at nonzero t we obtain a non-zero contribution from the 3-bare-particle state $a^{\dag}b^{\dag}c^{\dag} |0 \rangle$. Three new particles are created spontaneously from the vacuum. I consider this a sign of a serious trouble, because nobody has ever seen experimentally creation of extra particles from the vacuum and 1-particle states. I believe that vacuum and 1-particle states must be eigenvectors of the full Hamiltonian. Only "dressed" vacuum and 1-particle states satisfy this requirement.

My problem is that I'd prefer to deepen my knowledge of standard QFT before I delve into non-standard works with about 400 pages...

So genneth told me that it is well accepted that the states we primarily describe in theory are not the ones we measure. What key words do I have to search for or, more specifically, which chapter do I have to read in one of the standard textbooks in order to understand this better ?

Unfortunately, I haven't seen a clear explanation of the problems with bare particles in QFT in textbooks. In standard QFT, where you are interested only in the scattering (S-)matrix, the unphysical character of bare states is not a pressing issue, and one can perform sensible calculations without paying attention to such "subtleties". By the way, the 400+ pages that you mentioned were written for readers who don't have any previous knowledge of QFT, so they may be less intimidating than they look.

Eugene.

 

[OOO]

$$|0(t) \rangle = \exp(\frac{i}{\hbar}Ht) |0 \rangle \approx (1+ \frac{it}{\hbar}(H_0 + a^{\dag}b^{\dag}c^{\dag}) |0 \rangle = |0 \rangle + \frac{it}{\hbar} a^{\dag}b^{\dag}c^{\dag} |0 \rangle$$

So, at nonzero t we obtain a non-zero contribution from the 3-bare-particle state $a^{\dag}b^{\dag}c^{\dag} |0 \rangle$. Three new particles are created spontaneously from the vacuum. I consider this a sign of a serious trouble, because nobody has ever seen experimentally creation of extra particles from the vacuum and 1-particle states. I believe that vacuum and 1-particle states must be eigenvectors of the full Hamiltonian.

That confuses me since the vacuum is defined as the lowest eigenstate of the Hamiltonian. So your claim amounts to saying that either you are working with the false vacuum state or the Hamiltonian of QED does not possesses a lowest eigenstate.

By the way, the 400+ pages that you mentioned were written for readers who don't have any previous knowledge of QFT, so they may be less intimidating than they look.

I acknowledge that. Nevertheless you can't tell what you need and what you don't by looking at the front cover.

 

[meopemuk]

That confuses me since the vacuum is defined as the lowest eigenstate of the Hamiltonian. So your claim amounts to saying that either you are working with the false vacuum state or the Hamiltonian of QED does not possesses a lowest eigenstate.

The bare vacuum state (which I denoted $|0 \rangle$) is the lowest (zero) energy eigenstate of the free Hamiltonian $H_0$. However, as you correctly pointed out, it is not an eigenstate (lowest energy or otherwise) of the full interacting Hamiltonian $H$. Actually, traditional QFT doesn't bother to find the true vacuum state (the eigenstate of $H$). In order to calculate the S-matrix elements (which is the major goal of the traditional QFT)

$$S_{if} = \langle 0 | abc \ldots S \ldots c^{\dag} b^{\dag} a^{\dag} | 0 \rangle$$

it is sufficient to know the bare vacuum $| 0 \rangle$, bare particle creation and annihilation operators $a, a^{\dag}, \ldots$, and the expression of the scattering operator $S$ through these bare particle operators.

Eugene.

 

[Maxwell's Beard]

It doesn't sound like you're talking about classical aetherists at all! They were rejecting the notion of an underlying field, and instead asserting that the electromagnetic field describes the bulk properties of an underlying material.

e.g. the stereotypical line "How can you have a wave if there isn't anything for it to wave through?"

I write like I talk I guess so a lot is lost if we aren't in the same room together. I have said numerous times that these guys were wrong, wrong, wrong in their conclusions and definitions. My assertion is that they may not have been in the right ballpark, but they were in the right league. The aetherists believed that even a vacuum wasn't empty, the quantum field theorists posit the same.

 

[meopemuk]

Maxwell's Beard,

I think that fundamentally you are right. Modern physicists tend to ascribe to the empty space (vacuum) properties of some material substance/medium/continuum/... However, they will swear that this has nothing to do with the long discredited aether. You should take these statements with healthy scepticism.

The vacuum with non-trivial properties appears in different sections of modern physics. We already discussed the "vacuum polarization" in basic QFT. Those studying "spontaneous symmetry breaking" in quantum gauge theories may tell you that physical vacuum is "like superconductor". In general relativity it is assumed that the space-time has certain properties, including curvature, momentum-energy, etc. Many people believe that at the Planck scale one would find some non-trivial space-time structures, like discreteness, non-commutativity, and who knows what...

All these models of vacuum are extremely complicated, and what is worse, they have no direct experimental manifestations. In these respects, the similarities with the 19th century aether are striking. You are certainly right about that. In my opinion, the weapon against "physical vacuum" should be the same as Einstein used in 1905 against the aether - focus on experimental measurements.

Eugene.

 

[Hurkyl]

I write like I talk I guess so a lot is lost if we aren't in the same room together. I have said numerous times that these guys were wrong, wrong, wrong in their conclusions and definitions. My assertion is that they may not have been in the right ballpark, but they were in the right league. The aetherists believed that even a vacuum wasn't empty, the quantum field theorists posit the same.

That's an incredible stretch. As far as comparisons go, it doesn't seem much better than saying that anyone who thinks that the universe contains stuff with properties has an intuitive grasp of {insert any scientific theory here}.

 

[OOO]

The bare vacuum state (which I denoted $|0 \rangle$) is the lowest (zero) energy eigenstate of the free Hamiltonian $H_0$. However, as you correctly pointed out, it is not an eigenstate (lowest energy or otherwise) of the full interacting Hamiltonian $H$. Actually, traditional QFT doesn't bother to find the true vacuum state (the eigenstate of $H$). In order to calculate the S-matrix elements (which is the major goal of the traditional QFT)

$$S_{if} = \langle 0 | abc \ldots S \ldots c^{\dag} b^{\dag} a^{\dag} | 0 \rangle$$

it is sufficient to know the bare vacuum $| 0 \rangle$, bare particle creation and annihilation operators $a, a^{\dag}, \ldots$, and the expression of the scattering operator $S$ through these bare particle operators.

Eugene.

As I have indicated I haven't cared much about perturbation theory yet. But one thing I tentatively assume: If one uses the vacuum of the free Hamiltonian one is not to expect that it is also the vacuum of the coupled Hamiltonian.

Thus I cannot see any contradiction in what you have said:

$$|0(t) \rangle = \exp(\frac{i}{\hbar}Ht) |0 \rangle \approx (1+ \frac{it}{\hbar}(H_0 + a^{\dag}b^{\dag}c^{\dag}) |0 \rangle = |0 \rangle + \frac{it}{\hbar} a^{\dag}b^{\dag}c^{\dag} |0 \rangle$$

So, at nonzero t we obtain a non-zero contribution from the 3-bare-particle state $a^{\dag}b^{\dag}c^{\dag} |0 \rangle$. Three new particles are created spontaneously from the vacuum. I consider this a sign of a serious trouble, because nobody has ever seen experimentally creation of extra particles from the vacuum and 1-particle states. I believe that vacuum and 1-particle states must be eigenvectors of the full Hamiltonian.

 

[meopemuk]

If one uses the vacuum of the free Hamiltonian one is not to expect that it is also the vacuum of the coupled Hamiltonian.

Why not? What is the reason to think that interaction should change the definition of vacuum and 1-particle states? The word "inter-action" itself suggests that this is something present when there are two or more particles. Zero-particle and 1-particle states should not care whether there is interaction or not. This means that interaction Hamiltonians should yield zero when acting on such states, and this is the main postulate of the "dressed particle" approach.

If you assume that interaction is present also in the vacuum and 1-particle states, i.e., that "bare" and "physical" vacuum and particles are different, then you'll need to deal with ugly problems of "vacuum polarization", "self-energies", renormalization, etc. All these problems don't have any physical meaning, in my opinion, and they are absent in the "dressed particle" approach.

Eugene.

 

[OOO]

Why not?

Well, because you said so. In the equation

$$H = H_0 + a^{\dag}b^{\dag}c^{\dag} + \ldots$$

H and H0 can clearly only have the same vacuum state (with the same vacuum energy) if

$$(a^{\dag}b^{\dag}c^{\dag} \ldots)|0\rangle = 0$$

but then these terms do not create any particles from the vacuum.

 

[meopemuk]

Well, because you said so. In the equation

$$H = H_0 + a^{\dag}b^{\dag}c^{\dag} + \ldots$$

H and H0 can clearly only have the same vacuum state (with the same vacuum energy) if

$$(a^{\dag}b^{\dag}c^{\dag} \ldots)|0\rangle = 0$$

but then these terms do not create any particles.

Here is an example of the Hamiltonian, which, in my opinion, is entirely satisfactory

$$H = H_0 + V$$
$$V = a^{\dag}b^{\dag}ab + a^{\dag}b^{\dag}c^{\dag}ab + a^{\dag}b^{\dag}abc$$...(1)

Clearly

$$V | 0 \rangle = 0$$
$$V a^{\dag}| 0 \rangle = 0$$
$$V b^{\dag}| 0 \rangle = 0$$
$$V c^{\dag}| 0 \rangle = 0$$

so the interaction does not act in the vacuum and 1-particle states. The first term in (1) describes direct interaction between particles a and b. The second term describes emission of particle c in a+b collisions. The third term in (1) describes absorption of particle c in a+b collisions. More complicated particle-number-conserving and non-conserving terms can be added to (1), if needed.

It is however important that one should avoid terms of the type $a^{\dag}b^{\dag}c^{\dag}$, $a^{\dag}c^{\dag}a$, etc. which have non-trivial actions on 0-particle and 1-particle states. There is no experimental evidence that particles can be spontaneously created from vacuum and (stable) 1-particle states. So, the presence of such terms in the Hamiltonian is not welcome.

Eugene.

 

[gamburch]

Having one's aether and living in it too.

The following is an exerpt from a paper I have recently written on this very subject. It's a little long, but if you are really interested I can e-mail a copy of the whole thing and would like to get your feedback. I believe there is an aether, but not of the kind considered to date.

Albert Einstein described a treatment of physics consistent with the results of Michelson and Morley in the Special Theory of Relativity. In this theory all observers in motion with respect to each other, surprisingly, measure the same rules for all physical laws in their own frames of reference. This democracy among observers in motion is inconsistent with the concept of a three-dimensional aether. With a three-dimensional aether it is to be expected that different motions with respect to the aether will result in measurably different effects upon the propagation of light and other physical phenomena as seen by different observers, each in their own frames of reference. Einstein's Special Theory, however, represents there are no such measurable effects arising from motions by different observers with respect to any supposed aether thus contradicting the properties of the aethers envisioned at the time. The Special Theory has been shown to be in accord with experiment in all respects tested to date. As a result physics of the 20th Century was, for the most part, bereft of aethers.

In spite of the experimental success of the Special Theory, however, some of the epistemological beauty of physics has been lost by its validation. With Special Relativity all observers see surrounding space as stationary with respect to them. In other systems, for example, a sound source moving through air, a disturbance takes place with respect to the surrounding medium. Cause and effect are easily and readily imagined and measured. With Relativity, however, different observers, irrespective of their motions, see no movement relative to the medium in which they move. Such thoughts suggest that the aether is either carried with the body or arises from some other rather mysterious cause that is not subject to normal analogy. The first option was long ago discredited by means of experiment. The state of present knowledge is the latter.

In this paper we propose a different way to address these matters and seek to describe physical systems in such a way that disturbances from moving objects propagate with respect to the aether in which the disturbances arise, thus rectifying common sense with physical theory. Yet our model supports Relativity Theory with the fact that observers in motion with respect to each other each appear to be stationary with respect to happenings in the space they observe. We propose the existence of a four-dimensional aether in a space with four spatial coordinates and a time parameter so that properties of the aether at each point may vary both in space and time; which “time” is assumed uniform for all parts of the aether and increasing at a constant rate. Disturbances in the aether may propagate with reference to the aether. Thus if a signal is initiated in the aether by either a stationary (with respect to the aether) or moving object, the signal propagates in the same way in the stationary aether. The object moving with respect to the aether has any disturbance it initiated move with respect to the object after the initiation since the disturbance is propagating in the stationary aether. This situation is like acoustics or waves in a river. The appealing aspect of this situation is that, in initiating a signal, an object operates upon something, in this case the aether. Such an interaction is local. The results do not depend upon the rate of motion of the object in the aether since the emission, by its nature, is local. One thus maintains a simple and clean concept of causality in the case of four-space aether discussed herein.

In this four-space let us conceive of a particle (as opposed to the object of the previous paragraph) as a point moving on a path in the four-space. The particle’s size in the three-space perpendicular to its direction of travel is assumed to be small. The path can be in any direction in four-space and, for Special Relativity, need not be curved. The particle travels along this path at the speed of wave propagation in the aether with time determining its position along the path. Since particles traveling more or less in the same direction at nearly the same time can be put together to make observers (all observers are assumed to be collections of particles), we shall consider that all observers are moving at the same rate as a particle, and this rate is the same as signal propagation in the aether.

Since observers communicate with each other with signals traveling at the same rate as their speed with respect to the aether, we must examine the effects of motion of each observer with respect to the aether and each other. Observers following paths in different directions of limited angle with respect to the each other appear to be moving with respect to each other. Since the aether is isotropic and all observers are moving at the same speed with respect to the aether, we should expect each to see the same situation as any other. Furthermore, since each observer is moving at the same speed as his means of communication in the four-space, the coordinate in his direction of motion is annihilated so that he observes a three-space. This three-space, with its fourth coordinate obscured by being in the direction of travel of the observer, can rotate in four-space to change its direction of motion with respect to other observers. By these means observers see the space of Special Relativity. We attempt to make these matters somewhat more clearly in the body of the paper using some simple diagrams.

 

[cyberdyno]

If the aether is physical but non-material, with no landmarks, how could it represent a fixed frame?

Einstein was right, the Universe is background free.

 

[cyberdyno]

Why would one give the name "aether" to the vacuum (of QM) when it already has a perfectly good name?

Because that is its rightful name, given to it over four thousand years ago.

 

[cyberdyno]

Therefore, if you are an early 20th C physicist and you believe that the "aether" is Newton's absolute frame, then yes, you are wrong.

Einstein, Minkowski and Mach described a different aether. This twentieth century aether differs from earlier aethers in that in it objects are relative to other objects, not to empty space, therefore, there is no POR violation.

Empty space = aether

 

[cyberdyno]

So I pose this question again: what is all that different about a "stressed aether" giving rise to electric and magnetic fields, compared to an "excited vacuum" giving rise to an electromagnetic field?

Like Einstein said, there is no absolute space, space is an extension of matter. Space is not primary, nor fundamental, it does not exist by itself, it is a product, just as matter and time are products. Space is dynamic, it fluctuates, it tells matter where to go and matter tells it how to curve, remember? Empty space, on the other hand, is primary. You see, there is empty space and then there is material space, a mix of ZPR and CMBR particles. Einstein' spacetime is packed full of photons, that is where Inflation, the time cone, the time arrow, the Big Bang, 'false vacuum', etc., all come from. This why we now say space is grainy.

Locality, in spacetime, is a relation. Objects are relative to other objects, not to empty space.

The field is not to be seen as the ultimate irreducible reality, empty space is. But information starts with the field... with first quantum of action.

When we think about empty space we should stay away from notions that imply motion. Terms like infinity or velocity, size or duration... are not applicable. In this realm, we must think in terms of state, not in terms of process. Process happens in spacetime.

Because the aether is not composed of parts that follow a time line and the idea of motion is not applicable, we can safely say that the aether is one. Because it is one, there is no need for motion, there is no space or distance to cover, this is where non-locality and EPR phenomena come from. State, not knowledge, is registered throughout the Universe instantaneously, Mach was right.

---------

"...in topological quantum field theory we cannot measure time in seconds, because there is no background metric available to let us count the passage of time! We can only keep track of topological change."

"The topology of spacetime is arbitrary and there is no background metric."

"Quantum topology is very technical, as anything involving mathematical physicists inevitably becomes. But if we stand back a moment, it should be perfectly obvious that differential topology and quantum theory must merge if we are to understand background-free quantum field theories. In physics that ignores general relativity, we treat space as a background on which the process of change occurs. But these are idealizations which we must overcome in a background-free theory. In fact, the concepts of 'space' and 'state' are two aspects of a unified whole, and likewise for the concepts of 'spacetime' and 'process'. It is a challenge, not just for mathematical physicists, but also for philosophers, to understand this more deeply." ------ John C. Baez (from "Higher-dimensional algebra and Planck scale physics", as it appeared on the book "Physics Meets Philosophy at the Planck Scale" by Craig Callender and Nick Hugget)

 

[gamburch]

Cyberdyno:

Perhaps my overly long discussion of the difference between a space with four real coordinates and a time parameter containing an aether that has inertial and elastic properties is a bit obscure. Please be assured I am trying to keep its properties to the simplest I believe relevant. I thus ask you to envision a space with four real spatial dimensions. In this space is an aether with properties such that waves can propagate or the aether can flow as time goes by as seen by creatures in the four-space (if such creatures exist). Further suppose such creatures can cause disturbances in the aether that propagate at the speed of light with respect to the aether. The situation is no more complicated than a man in a boat dropping stones in the water or an airplane traveling through air. The leap I'm asking you to take is that this aether can be described with four spatial dimensions and that changes in the aether take place with respect to a universal time parameter. Please note this has nothing to do with any concept (that I know of) that was discussed in depth by Einstein, Mach, of anyone else.

Let us now consider that this ather is observed from frames of reference that move through the aether at the speed of wave propagation in the aether. Presumably such observations can only be imagined if observers are composed of waves traveling in some direction in the aether. The question is: What does such an observer see? In our world a analogous situation might be that of an airplane traveling at the speed of sound that can only send and receive sonic waves. In the case of the airplace that answer is discussed, for example, in Morse and Feshbach (Methods of Theoretical Physics( MacGraw-Hill, Inc., New York, NY. 1953, p 168). In summary, the pilot only "sees" a two dimensional subspace.

In the case of four-space an observer sees other observers, moving in slightly different directions, as traveling with respect to him in a three space at speeds slow compared to the speed of wave propagation. As the angle increases between observers, each will note effects exactly like Special Relativity. Thus in this model, we are really creatures of a four dimensional space, which space need have properties normally associated with the kind of causality envisioned before Special Relativity. We observe Special Relativity not because it is fundemental, but because we observe the aether from a frame of reference moving at the speed of signal propagation in the aether.

I have no argument if you insist that Special Relativity "is" because it is. After all there is nothing more to say. What I seek is an explanation for why the logical appeal of causality does not seem to apply with Special Relativity. Here is an explanation, however outlandish it might seem at the outset.

I don't believe it appropriate to discuss such matters here, but the set of assumptions involved with the model I presented do imply the validity of Mach's principle, the value of inertial forces, the pleasing concept that such forces do indeed arise from all other matter, Generl Relativity, and classical mechanics. As you note the concepts are purely classical so that and quantum effects are not discussed.

 

[Hurkyl]

Because that is its rightful name, given to it over four thousand years ago.

Seeing how the theory of quantum physics wasn't known four thousand years ago, I have trouble believing that one of its technical terms was named during that era.

 

[russ_watters]

Reading the first few posts, I fail to understand how the thread got this far. The OP correctly notes that the the classical ether is different from Einstein's ether and different from the QM vacuum. I'd have preferred it if Einstein hadn't used the word "ether" in that speech, but then the word "plane" is also used to describe utterly different things as well, so how big of a deal should that really be? Just because the same word is used, that doesn't make the concepts identical. Kids learn that concept in elementary school when learning to read. There really isn't anything else to this beyond that.

[note: yes, I see that this is a resurrected thread]

 

[russ_watters]

Seeing how the theory of quantum physics wasn't known four thousand years ago, I have trouble believing that one of its technical terms was named during that era.

Put another way, just because someone selected a pre-existing word to describe a new concept, that doesn't mean the new concept is the same as a previous concept that was given the same name.

I'm quite certain I could make a 4 year old understand this. It is sooooo trivial.

 

[cyberdyno]

Reading the first few posts, I fail to understand how the thread got this far. The OP correctly notes that the the classical ether is different from Einstein's ether and different from the QM vacuum. I'd have preferred it if Einstein hadn't used the word "ether" in that speech, but then the word "plane" is also used to describe utterly different things as well, so how big of a deal should that really be? Just because the same word is used, that doesn't make the concepts identical. Kids learn that concept in elementary school when learning to read. There really isn't anything else to this beyond that.

[note: yes, I see that this is a resurrected thread]

This notion of a primordial substance is a very old one, also known as Akasha or Brahman, and many times described as pure energy or spiritual fire. It has been anthropomorphized by man since the times of Plato and Aristotle, the Chaldeans and the Akkadians. It has been called by the names of Zeus, Jupiter, Brahma and other.

We are talking about a notion, not just a word.