Let us begin with basic definitions. The magnetic field H is defined in terms of magnetization M and magnetic induction B (often called just magnetic field) via he formula H= B/μ0-M quite generally. For historical reasons different units are used for B and H but in vacuum they are essentially one and the same thing. In the case of magnets the situation changes.
Paramagnets and diamagnets are linear: M= χm H in good approximation. The sign of χm determines whether the behavior is diamagnetic (induced magnetization reduces the external field inside magnet) or whether its paramagnetic (induce magnetization increases the external field inside magnet). For the ferromagnets the behavior
is non-linear and M is non-vanishing for vanishing H.
The behavior at the boundary between magnet and external world is important for understanding magnets. If the exterior of the magnet is vacuum, one has H=μ0M and the fields H and B are one and the same thing. Inside magnet magnetization implies that the two fields are not same. By the absence of magnetic charges the normal component of magnetic field is continuous. Therefore the normal component satisfies Bn(in)= μ0Hin+M= Bn(out)=μ0Hout.
Hysteresis for ferromagnets
For ferromagnets the relationship between H M is non-linear and many-valued, and one has hysteris so that the effects of increasing and decreasing the magnetic field are different in saturation. The hysteresis cycle characterizes this behavior quite nicely.
- The plot can be understood easily when one realizes that magnetization M corresponds microscopically to the density of magnetic dipoles typically realized as electrons spins. The field H corresponds essentially to the difference between external magnetic field and that caused by magnetization. As one starts from H=0 to increase the magnitude of H, M gradually increases as for paramagnets but above some critical value M starts to increase and reaches the saturation value. The interpretation is that electron spins change they direction in phase transitions producing magnetized regions. This behavior can be understood only quantum mechanically and can be seen as an outcome of exchange interaction for spins relating to fermion statistics. Essentially collective behavior is in question: when the number of spins in direction of external field is large enough all other spins "follow the fashion".
- One can also gradually decrease the value of H in saturation. Now M decreases smoothly as some spins here and there change their direction but no phase transition like reduction of M occurs since "follow the fashion" behavior is possible only when the number of reversed spins is high enough. Therefore one proceeds along the lower part of the hysteresis cycle: inside the cycle the behavior is reversible for small enough changes of H and irreversibily occurs only near the ends.
- At some point one has a situation in which the added external field vanishines: H=0. There is a net magnetization and the normal component of external field is essentially this magnetization at the surface of the magnet. This field is known as remanescence. This corresponds to a magnetization without any added external field H.
- As H is reduced further so that its sign changes, eventually "follow the fashion" behavior sets in and most spins change their direction to opposite and the direction of magnetization changes. The magnitude of the magnetic field at which this occurs is known as coercivity. Coercivity and remanescence characterize the magnetic in the first approximation.
- The mathematical description of Higgs mechanism is very similar to the idealized description of spontaneous magnetization. In spontaneous magnetization one has however large number of magnetized regions which means that the direction of magnetization does not change instantaneously since the analog of overheating and overcooling are possible for all these regions.
Hysteresis (and hysteria) in sociology
The hysteresis model applies also in sociology. Superstring fashion represents an excellent example about hysteresis phenomenon in the sociology of science. There are two kinds of people involved: those who really understand (H) and those who just follow the fashion (M). Some fraction p of those who understand, believes in superstrings. The remaining fraction 1-p does not. H dould correspond mathematically to the difference (p-p0)/p0, where p0 is a critical propability analogous to remanesence in ferromagnetism. As p increases a sudden phase transition occurs at critical fraction satisfying p-p0=Δ pcr, and everyone suddenly believes that super strings are not only a promising theory but even the only possible one. Δ pcr is analogous to coercivity.
The return along upper part of hysteris curve means that the per cent of the believers who really understand starts to decrease as people who have not suffered the fate of becoming a string guru are learning about the deep problems of the theory and start to worry about them publicly. Eventually the number of the non-gurus who understand and do not believe becomes so high that the followers of fashion make the decision to not believe anymore on superstrings, and everyone suddenly "knows" that super strings are pseudoscience. Except certain billionaire who decides to make super string theorists millionaires;-) . This billionaire does not follow the fashion, which is nice. He does not even follow those who understand, which is not so nice. To my humble opinion, it would be more rational to build an institute supporting the study of alternatives for super string models than drowning theoreticians to money which is not even their primary interest.
Hysteresis in magnets cannot be understood microscopically without introducing quantum effects. One can of course ask, whether also hysteresis involves quantum mechanics but now in macroscopic length and time scales applying in sociology.
Ferromagnetism, catastrophe theory, and Higgs mechanism
The model for magnetism can be also understood in terms of catastrophe theory using cusp catastrophe. This applies to Higgs mechanism (not the manner in which particle massivation is described in TGD) too. Catastrophe theory brings in also the temperature as external parameter besides H. This is the simplest interesting catastrophe and involves two parameters/control variables: (temperature T and added external field H) plus one behavior variable (magnetization M). Above certain critical temperature (Curie temperature) there is no magnetization. Below Curie temperature there is a magnetization in certain range for field H. As the magnitude of H is increased, one reaches at critical magnitude of H (coercivity) a situation when the only possible manner to continue is to jump from the boundary of the cusp to the sheet below or above it. A sudden phase transition like change changing the sign of magnetization takes place.
Above we have assumed that the external contribution H to the field is in the same direction as magnetization. What happens if this is not the case? And what happens when the external field H changes the direction of magnetization: is it enough apply the basic criterion to the vector sum H= B/μ0-M. Presumably this is the case. If it were not, it would be mentioned in Wikipedia!
Catastrophe theory provides a rough description and does not say anything about the microscopic mechanism of hysteresis. In the case of Higgs mechanism one can expect the same to be true: there should exist a microscopic thermodynamical description replacing Higgs mechanism, which is only capable of reproducing the particle masses but not predicting them and suffers from un-naturality. In TGD framework p-adic thermodynamics provides this microscopic thermodynamical description.
Since QFT is in rough sense square root of thermodynamics, one can argue that also p-adic thermodynamics must have also square root and characterizes single particle states rather than ensembles. This is the case. In zero energy one can say that quantum theory is square root of thermodynamics. U-matrix characterizes the physics. I call the rows Mi of unitary U-matrix M-matrices, which are products of mutually orthogonal hermitian square roots Pi1/2 of density matrices Pi with a unitary S-matrix S. One has MiMj†= Piδij and Tr(MiMj†)=δij. U defines negentropic entanglement in time direction. In p-adic thermodynamics the Hermitian matrix Pi1/2 corresponds to a diagonal matrix defined by the square roots of eigenvalues of mass squared operator expressible as vibrational part of scaling generator L0 of conformal algebra. The unitary matrix S is the counterpart of the ordinary S-matrix.
Permanent magnet in many-sheeted space-time
How could one understand permanent magnet in the framework of many-sheeted space-time?
- It seems that one must consider two space-time sheets. That of magnet at which magnetization M alone defines the magnetic field B1=μ0M and that of external world at which external added field defines the magnetic field B2=μ0 Hin. Test particle has topological sum contacts on both space-time sheets and effectively experiences the sum B1+B2: the fields do not however sum at same space-time sheet - only their effects on spins are additive so that '+' is replaced with a set theoretic union inTGD. This applies quite generally and saves TGD from the catastrophe caused otherwise by the fact that linear superposition of fields is not possible at given space-time sheet even approximately.
- The spins inside the space-time sheet of magnet also experience B2 and when B2 is above critical value (coercivity), a phase transition changing the direction of spins occurs and magnetization is generated. It seems that TGD does not add much to the model of how spontaneous magnetization takes place and what is behind the hysteresis.
- The notion of magnetic field in TGD differs from that in Maxwell's theory. In TGD Universe it is quite possible that the flux tubes have closed cross section (sphere) instead of disk and that they carry a monopole flux. One would have magnetic flux tube without a circulating current at its boundary generating the magnetic field inside it as in the case of electromagnets, which are typically flux tubes with current carrying helical wire. Could the static magnetic field of permanent magnet consist of monopole flux tubes with a closed rather than disk-like cross section requiring no circular currents at their boundaries?
If this kind of flux tubes are possible, one could understand why cosmos is populated by magnetic fields in all scales. In Maxwell's theory this is not possible since coherent currents defining flux tubes are not possible in primordial cosmology. In superconductors the flux quanta associated with this monopole like fields would become visible as they penerated the super-conductor.