  Five TeV neutrino events weakly correlated with five gamma-ray bursts (GRBs)
were detected recently by IceCube. This work is an attempt to show that if the
GRB identifications are verified, the observed time delays between the TeV
neutrinos and gamma-ray photons from GRBs provide attractive candidates for
testing fundamental physics with high accuracy. Based on the assumed
associations between the TeV neutrinos and GRBs, we find that the limiting
velocity of the neutrinos is equal to that of photons to an accuracy of
$\sim1.9\times10^{-15}-2.5\times10^{-18}$, which is about $10^{4}-10^{7}$ times
better than the constraint obtained with the neutrino possibly from a blazar
flare. In addition, we set the most stringent limits up to date on the energy
scale of quantum gravity for both the linear and quadratic violations of
Lorentz invariance, namely $E_{\rm QG, 1}>6.3\times10^{18}-1.5\times10^{21}$
GeV and $E_{\rm QG, 2}>2.0\times10^{11}-4.2\times10^{12}$ GeV, which are
essentially as good as or are an improvement of one order of magnitude over the
results previously obtained by the GeV photons of GRB 090510 and the PeV
neutrino from a blazar flare. Assuming that the Shapiro time delay is caused by
the gravitational potential of the Laniakea supercluster of galaxies, we also
place the tightest limits to date on Einstein's weak equivalence principle
through the relative differential variations of the parameterized
post-Newtonian parameter $\gamma$ values for two different species of particles
(i.e., neutrinos and photons), yielding $\Delta\gamma \sim 10^{-11}-10^{-13}$.
However, it should be emphasized again that these limits here obtained are at
best forecast of what could be achieved if the GRB/neutrino correlations would
be finally confirmed.
