## Are there infinities in nature?

Infinity is a mathematical quantity - no physical quantity is ever infinite.

The concept of infinity is endlessly fascinating. For philosophers, for mathematicians and of course children. We have all wondered about how any number can be infinite, because we can always imagine making a larger number: infinity plus one. The concept of infinity is also discussed in the context of the physical sciences. However, frequently people neglect to mention that there are no infinite quantities in physics. None? Well, I don’t think so. I think that infinity is fundamentally a mathematical concept and in reality something else always happens that interferes with the extremity of physics that would happen near an infinite quantity of anything.

Are you sure? Well ‘No’.

I agree that in the physical sciences, the word infinity is often used. I remember being taught at school that when one’s eyes were relaxed they were ‘focussed on infinity’. I wondered about that phrase for a long time, but in the end I concluded it was just poetry. It means one’s eyes were set so that parallel light would be focussed onto one’s retina. If the light had indeed come from infinity,  it would – of course – not have reached us yet.

What about the singularity around an electron?  At Sussex University, Dr David Bailin taught me that a ‘bare’ electron would have infinite mass and infinite electrical charge. I was told that the particle that we observe and call ‘an electron’  is actually a hypothetical ‘bare electron’ plus its ‘re-normalised cloud of virtual electrons and positrons’. This cloud is created by the intense electric field around the infinite electrical charge. What that means is that in reality we never observe an electron to have infinite mass, but theoretical physicists imagine a real electron as being composed of a hypothetical singular particle plus another phenomenon that hides the singularity. The upshot of this is that a ‘bare electron’ is a mathematical concept – not a physical one. When we look at electrons we never observe an infinite property.

Similarly, it is populalry stated that a black hole is a ‘gravitational singularity’ – an infinitely strong peak of gravitational intensity. But of course we have no observations on this and based on everything we know, we would expect that as the field intensity increased – ‘something’ would happen. Currently, we have no idea what that ‘something’ is. But there is certainly no experimental evidence that a gravitational singularity actually exists. The intensity can reach any amazingly large number. It can reach an intensity which is so alarming that it takes my breath away. But as long as a number can be associated with it – it is not infinite. I am prepared to be ‘boggled’ by large numbers but not ‘baffled’ by an unphysical concept.

What about the Big Bang? At the time of the Big Bang – something is supposed to have exploded for reasons we have not yet figured out. Amazingly it seems that the vast universe hat we observe, and all the energy in it, was once packed closely together in the space occupied by a proton. So the universe was once unimaginably hot and dense. Do I mean infinitely hot and dense? No, not infinitely so, just orders of magnitude beyond any regular conception of temperature or density.

What about the Universe? Well there are many different conceptions of the Universe – and we have data from the cosmic microwave background indicating that the furthest structures that we can see are at most around 40 billion light years distant. Beyond that there is certainly something else, but we just don’t know what. But surely, I hear you ask, one can always keep going? Well actually, we just don’t know! In some conceptions one can, and in others one can’t. But there is no reason to suppose that the Universe is infinite – only that it is even more uncomfortably large than we previously conceived.

So is infinity unphysical? Well I think so. I would love to hear of an example of an infinite quantity, but actually I just don’t think that an infinite ‘anything’ makes any sense at all. And as a measurement scientist I would be very interested to know of the uncertainty of measurement of an infinite quantity: infinity plus or minus what?

### 6 Responses to “Are there infinities in nature?”

1. David Bailin Says:

The infinity in QED, and other quantum field theories, derives from a short-distance cut-off which should be zero, IF there is no new physics at all. We know that QED does not include quantum gravitational effects, and the electron surely interacts with gravity. So at the very least there should be a cut-off at the Planck length, corresponding to an energy scale of 10^{19} GeV. The observed (finite) mass is then the sum of the bare mass and the finite quantum radiative correction, so evidently the bare mass is finite too. I share your (and Dirac’s) view that infinity in a phsical theory is a sign that something is wrong, i.e. that there is other relevant, generally new, physics.
Best wishes
David Bailin

2. David Bailin Says:

Further, the radiative correction to the electron’s mass, and in general any fermion’s, depends only logarithmically on the cut-off. Even if the cut-off is as large as the Planck scale, the radiative correction is of the same order of magnitude as the observed mass, and so, therefore, is the bare mass.
Now suppose that the Higgs boson is indeed discovered at the LHC, with a mass of order 125 Gev/c^2. Unlike a fermion, the radiative correction to the mass(-squared) of a SCALAR particle is proportional to the cut-off (squared). In this case then, if the cut-off is of order the Planck scale, the bare mass(-squared) must also be of this order; in fact the bare mass-squared and the radiative correction have to cancel to one part in 10^{34}! Many people, including me, think that this is implausible, although there is no theoretical reason why it should not be thus. It is an aesthetic objection called the “fine-tuning problem”. The only known way to evade it is SUPERSYMMETRY, which requires all of the known particles to have partners with the opposite statistics: fermions have (scalar) boson partners (selectrons, squarks, sleptons), and bosons have fermionic partners (photinos, gluinos, Winos, Zinos, Higgsinos). None have yet been observed, so the symmetry cannot be exact. However, the breaking cannot be at too high a scale. Otherwise it will not solve the fine tuning problem. This is why we expect SUSY to be discovered soon. It is another reason why the discovery of the Higgs would be of such importance. It would be the first known fundamental scalar particle, and would show that there is no obstruction in principle to the existence of the susy scalar particles.

3. Infinities in nature 2: SUSY, Squarks and Sleptons are the answer! « Protons for Breakfast Blog Says:

[...] Protons for Breakfast Blog Making sense of science « Are there infinities in nature? [...]

4. Katie Keith Says:

I agree, talking about infinity in these terms is something that has always annoyed me – particuarly with BBC documentaries! How can there be an ‘infinite number of planets’ or a singularity with ‘infinite mass’? Just because a number can be infinite doesn’t mean that something physical can too.

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