What is the conserved quantity of a scale-invariant universe?

Question

Consider that we have a system described by a wavefunction $\psi(x)$. We then make an exact copy of the system, and anything associated with it, (including the inner cogs and gears of the elementary paticles, if any, aswell as the fabric of spacetime), but where all distances are multiplied by a number $k$, so $\psi(x) \to \psi(kx)$, we consider the case $k>1$ (if $k=-1$ this is just the parity operation, so for $k<0$ from the little I read about this we could express it as a product of P and "k" transformations).

Consider then that all observables associated with the new system are identical to the original, i.e. we find that that the laws of the universe are invariant to a scale transformation $x\to kx$.

According to Noether's theorem then, there will be a conserved quantity associated with this symmetry.

My question is: what would this conserved quantity be?

Edit: An incomplete discussion regarding the existence of this symmetry is mentioned here: What if the size of the Universe doubled?

Edit2: I like the answers, but I am missing the answer for NRQM!

Answer 1

The symmetry you are asking about is usually called a scale transformation or dilation and it, along with Poincare transformations and conformal transformations is part of the group of conformal isometries of Minkowski space. In a large class of theories one can construct an "improved" energy-momentum tensor $\theta^{\mu \nu}$ such that the Noether current corresponding to scale transformations is given by $s^\mu=x_\nu \theta^{\mu \nu}$. The spatial integral of the time component of $s^\mu$ is the conserved charge. Clearly $\partial_\mu s^\mu = \theta^\mu_\mu$ so the conservation of $s^\mu$ is equivalent to the vanishing of the trace of the energy-momentum tensor. It should be noted that most quantum field theories are not invariant under scale and conformal transformations. Those that are are called conformal field theories and they have been studied in great detail in connection with phase transitions (where the theory becomes scale invariant at the transition point), string theory (the two-dimensional theory on the string world-sheet is a CFT) and some parts of mathematics (the study of Vertex Operator Algebras is the study of a particular kind of CFT).

Answer 2

thank you for the nice question. It directly relates to the topics of conformal field theories. I found a very nice thread in another forum where I guess your question has been answered.
Nevertheless, I will try to summarize the main points here and maybe add some points.

Symmetries in General Relativity

In general relativities, symmetries correspond to an isometry of the metric $g=g_{ab}dx^a dx^b$, say $\varphi^\star g = g$. That means, if you move along the path of such a symmetry, it does not change. This can be expressed in terms of the Lie-derivative.
$$L_v g = 0$$ or $$\nabla_{(a}v_{b)} = 0$$

where the parenthesis stands for symmetrization over indices and $v = \dot\varphi(t)$ is the vector field associated to $\varphi$. One can find very nice introductory calculations for this in Robert M. Wald: General Relativity and Hans Stephani's Introduction to Special and General Relativity.

If $n$ is a unit geodesic, further integration of

$$Q=v_a n^a$$

leads to conserved quantities since $$n^a \nabla_a \left(Q = n^b v_b\right) = n^a n^b\nabla_b v_a + v_b n^a \nabla_a n^b\equiv 0 $$ due to the Killing and geodesic equations.

Famous examples are mass $M$ (or energy) for a stationary spacetime or angular momentum $J$ for axial symmetry (yes, one can assign a spacetime an angular momentum, I found it puzzling in the first place), $$M = 2\int_\Sigma \left( T_{ab}-\frac12 T^n_{\,n}g_{ab} \right)n^a \xi^b dV$$ $$J = -\int_\Sigma T_{ab}n^a \eta^b dV$$ where now $\xi$ is the stationary Killing vector, often $\xi = \partial_t$ and $\eta$, often $\eta=\partial_\varphi$ holds for the axial symmetry and $n$ is now vector perpendicular to a space-like hypersurface $\Sigma$.

Conformal isometries

Now, the situation is a little bit different. A conformal Killing vector $c$ now gives rise to a symmetry of the form $$L_cg=\omega^2g$$ and the conformal Killing equation, implicitly defining $\omega$ now takes the form $$\nabla_{(a}c_{b)} = \frac1n g_{ab}\nabla_d c^d$$

In your case, you force $\omega = 1$ but this is not of great importance as you will see next.

What happens to the "conservation equation"? We have $$n^a \nabla_a \left( n^b c_b\right) = \frac1n \left( \nabla_d c^d \right) n^a n_a$$

which is only zero if $n^a n_a = 0$, a null-geodesic. So, only for a very special class of movements, here light-particles, one will find a symmetry. But this was expected since conformal transformations will not change angles thus light movement won't be affected.

I don't think that this is a conserved quantity in the sense of Emmy Noether.

Sincerely

Robert

PS.: I apologize for any inconvenience concerning notation. I hope everything is clear from context.

Answer 3

Jeff Harvey has of course provided you with the perfect, standardized answer: the scale invariance boils down to the tracelessness of the stress-energy tensor. But the tracelessness is not really a "conserved quantity" in the usual sense that you may have waited for.

However, one may transform the problem to something that is a conserved quantity in the usual sense.

In particular, you may take your scale-invariant universe and insert a point-like object at a chosen point that I will call the origin. In quantum field theory, this is achieved by acting on the vacuum state with a local operator at the origin.

The transformations proving scale invariance are just radial expansions that keep the origin untouched. The laws of physics are invariant under these transformations, by assumption, and this symmetry is equivalent to the conservation of the dimension of the operator from the previous paragraph. But its conservation not with respect to the normal evolution in time but evolution in the "radial time", $\ln(r)$. Consequently, the dimensions of all operators are well-defined in scale-invariant theories. In scale-non-invariant theories, they would depend on the renormalization scale.

I added this verbal exercise in order to emphasize that the scale transformations in a scale-invariant theory are analogous - and in a very well-defined mathematical sense, equivalent - to ordinary translations in time. To be a bit specific, think about 2-dimensional Euclidean theories. The complex coordinate $z$ may be written as $\exp(a+ib)$. Here, $b$ is a periodic, angular variable with periodicity $2\pi$. However, $a$ is real and goes from $-\infty$ to $+\infty$.

The scale transformations are nothing else than the ordinary translations in $a$ which are linked to a Hamiltonian. For example, you expand $z$ $e$-times by shifting $a$ by one. And indeed, scale invariance in 2 dimensions implies the full conformal invariance - under all transformations that preserve the angles - so instead of looking at the $z=x+iy$ plane, you may equally well look at the $a+ib$ plane where the original scale transformation looks like an ordinary translation in the $a$ direction. By conformal symmetry, the form of the action in the $z$ and $a+ib$ coordinates are identical.

In higher dimensions, it is not quite true that scale invariance (and Lorentz/rotational symmetry) implies the full conformal symmetry, but in the important cases, it is true, anyway.

Best wishes Lubos

Answer 4

It's a standard result in the theory of fractals that any set of contraction mappings that don't overlap "too much" will have a unique attractor, and furthermore in principle these attractors have some Hausdorff dimension; I think this is the invariant quantity you are looking for. See for example Shakarchi and Stein, Volume 3, Chapter 7, Theorem 2.9.

Answer 5

Strict short answer to the Question – the number of particles is invariant (at large).
It is know that SM does not preserve energy, i.e. Noether is only valid as long as the ratio matter/space is constant.
From the above Answers we see that you don’t know of any scale-invariant theory that supports the physical laws.
The main question is: How to show that physical laws hold in a scale-invariant model? Many physicists tried and failed (Dirac, Canuto Hoyle and Narlikar , Maeder and Bouvier, Wesson).
I will present a resume of ‘A self-similar model of the Universe unveils the nature of dark energy’ by Alfredo G. Oliveira, submitted to PRX in July 1,2011. (Oh my gosh, my name is in the paper!)
If we attach a referential to a particle, say an atom, above represented in grey, we cannot find any evolution. It is our actual situation; we look around in the labs and we are naturally blinded to any evolution.
The Question mentions only a modification of the length, Lubos answer also mentioned a varying time, but that procedure is short of what is needed to have a correct self-similar model. It has to be done in a ‘physical’ way:
Lets shrink an atom (the atom is our reference for Mass, Length, Time) of the Past into the one of Present. As viewed from an external invariant reference ‘S’(Space) the Length Unit changed, and also the Mass Unit changed and also the Time unit changed because the speed of light is the constant c — it’s a property of the field/space.
It is evident that an atomic observer (bounded to his atomic reference) sees a space expansion. The cosmologic redshift of the light of galaxies (far away in time and distance) trace the fact that the atomic processes were slower in the past as compared to the ones of the present.
Be $M(t_S)=Q(t_S)=L(t_S)=T(t_S)=\alpha(t_S)$ the relation that describe the evolution of units thru time, as viewed by S in relation to the units of the atomic observer ( $\alpha(t_S)$ is the scaling law).
It is derived in the paper, using only the laws of physics, making no hypotheses, and departing only from measured data that the scaling law is $\alpha(t_S) = e^{-H_0\cdot t_S}$.
Quoting the Summary and Conclusions

Henry Poincaré analyzed how we acquire information, stressing the relative nature of our data and that our choice of units serves the convenience of obtaining the simplest form for physical laws;
Einstein analyzed how we calibrate reference frames, how we attribute coordinates to occurrences, what is the kind of time and length units we use;
here, the reflection on this subject is extended to the properties of the units, which enabled us to understand that the invariance of particles in standard units is a property of these units and not of the particles; it become also clear how the space expansion may trace a self-similar phenomenon and an important yet previously unnoticed property of the units of field constants was found, which is able of supporting the observed space dilation. From two accepted observational results, the invariance of constants and the scalar space expansion, and considering that the observed space expansion is consequence of a self-similar phenomenon, it is deducted a model that verifies the classic cosmic tests as well as the $\Lambda$CDM model in spite of having just one parameter, the Hubble parameter. This model has surprising features, namely:
(1) There is no theoretical conflict with fundamental physical laws but for a new term in one conser- vation law, which is beyond present possibilities of direct measurement.
(2) The standard systems of units lose their privileged role, physical laws being valid also in a space, comoving, system of units.
(3) In standard units, this model supports the same description of the universe of the $\Lambda$CDM model. Despite this scaling model not being a cosmological model, it gives some contributions to cosmology, namely:
(1) Space is older than matter.
(2) Matter, field and radiation evanesce in space units.
(3) A simple explanation arises for the lack of tendency for gravitational collapse.
(4) The roles of dark energy and of cosmological inflation are made clear.
This paper is just the first of a set of three; the second paper analyses the consequences of this model at the solar system scale and the third analyses the large-scale structure of the universe.
Until now, the knowledge of the universe was established in units where atomic properties are invariant; these units are very convenient for describing systems of bodies but, when used to describe space properties, the result is puzzling. To have surpassed this limitation is a major achievement of this work.

Of course one can argue 'I do not believe that it can happen to atoms!' and I will counter argue 'How can the space expand?'.
The paper is available here (the arxiv is closed to my friend Alfredo, probably not even Perelman can use arxiv anymore). I know this model since 1991, and a preliminary version can be found in the arxiv dated 2002; back then the public was unprepared to read this model and it is my expectation that we have evolved to a more mature position.