Monday, February 20, 2006

anthropic principle from a quantum field theory point of view

anthropic principle says that some foundamental parameters of the theory describing the universe, like the cosmological constant which is unnaturally small, are fixed without any foundamental reason, the reason is simply that otherwise human being cannot exist in such an universe. weinberg \cite{weinberg} first pointed out that existence of galaxies requires a small cosmological constant which is more or less the observed value. the electro-weak scale can be also understood by requiring stability of the proton and existence of the carbon \cite{sm}.

the natural understanding of the above principle is from the multi-verse and inflation. originally the whole universe contain many regions in which the foundamental parameters take different values. the part we are living (our universe) originates from a small region in which the parameters take their measured values. this region becomes large due to inflation. the string theory predicts \cite{bp} about 10^{500} regions which are enough to include a region like ours.

although the anthropic principle is hard to be proved experimentally, it maybe still true. if lhc mechine sees nothing beyond the standard model, personally i will believe this principle playing a role for the electro-weak scale.

to me, it is important to ask which parameters scan. it is not satisfactory if all the parameters scan. for an example, it is hard to take the yukawa couplings scan. although the small electron mass can be understood anthropically, the muon mass are hard to be understood in the same way, which is however in the equal footing as the electron mass.

i would like to look at the problem from quantum field theory of views. i consider qed with an elementary higgs. i simply assume that this field theory is the most foundamental one, and beyond which is the string theory or some quantum gravity theory. (the electron mass is taken as an example, which can be replaced by the electron yukawa coupling.)

let us look at quantum corrections to the parameters in the following,
m_e = (m_e)_0 \left( 1 + \alpha \ln \frac{\Lambda_{pl}}{m_e} \right) \\[3mm]
m_h^2 = (m_h)_0^2 + \Lambda_{pl}^2 \,,
where $\Lambda_{pl}$ is the cut-off. A cut-off used to be taken infinitely large, so the bare parameters were also infinitely large. however, the physical cut-off is always finite, therefore the bare parameters are also finite! because cut-off dependence of $m_e$ is logrithmic, in fact, the physical value and the bare value are very close, $m_e \sim (m_e)_0$. whereas for the higgs mass, the physical value and the bare value are very different.

accepting small $m_h^2$ from anthropic point of view, i take bare parameters to be physical. small $m_h^2$ means large $(m_h)_0^2$. A large $(m_h)_0^2$ is just natural (in the dirac sense). in other words, it is the difference
$(m_h)_0^2 + \Lambda_{pl}^2$ which scans. in this understanding the bare parameters are not require to scan in a wide range. if this is true, small bare parameter $(m_e)_0$ needs another mechanism to be understood.

my conjecture is that only the parameters which are finely-tuned can be anthropically determined. the other small parameters should be understood from certain symmetry breakings.

{weinberg} s. weinberg, phys. rev. lett. (1987)

{sm} phys. rev. d (1998) .

{bp} jhep.


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