Spontaneous symmetry breaking simply refers to the fact that the solution of a problem does not necessarily have the symmetry of the problem. Natural laws have some symmetries, e.g. Lorentz symmetry or translation invariance. Still, a chair or a particle are not translation invariant; nature allows for objects which do not have the symmetry of the laws that govern them. The equation x^2=1 is invariant under a change of sign x --> -x, but a specific solution like x=1 does not have that symmetry.

It certainly does not sound scientifically proper to call something spontaneous for it shuts out the door to investigation behind its causes. Or else it is just an adhoc proposal which is there just because it works. In my view, SSB must follow from some higher level underlying theory like supergravity or string theory or from some extension of the standard model.

Spontaneous symmetry breaking simply refers to the fact that the solution of a problem does not necessarily have the symmetry of the problem. Natural laws have some symmetries, e.g. Lorentz symmetry or translation invariance. Still, a chair or a particle are not translation invariant; nature allows for objects which do not have the symmetry of the laws that govern them. The equation x^2=1 is invariant under a change of sign x --> -x, but a specific solution like x=1 does not have that symmetry.

Dear Andreas,

Yours is the standard explanation that we find.

The question is about why and how does nature select the particular solution that we find in our universe. Mark your words, "Nature allows..." . Is it not the same as Spontaneity? It seems to be just another way of rephrasing "spontaneity". In many cases we do this habitually and are fooled into believing that we are explaining, till someone comes in with some fresh idea.

Thank you.

The explanation given by Andreas is actually correct that is why you call it "the standard explanation".

At the technical level the spontaneous symmetry breaking means just that the lowest energy configuration, the vacuum, does not have the symmetry of the Lagrangian. A particular vacuum to be realised in nature is chosen spontaneously. In the sense, that there can be another vacuum configuration chosen.

You can visualise this by thinking of a stick made of resin that stands vertically. This system is symmetric under rotations around the stick. Now if you apply vertical force to the upper end of the stick, it will deform to any direction. And on the level of mechanical theory it chooses the direction spontaneously. And the symmetry becomes broken since you no longer can rotate the deformed stick as it has a preferred direction.

There are very many instances of spontaneous symmetry breaking in nature, and all have an ultimate physical explanation - for instance, a homogeneous supercooled liquid will suddenly initiate a very fast phase change from some unpredictable point, but that point will happen have some characteristic - micro-unsmoothness, etc. - which makes it the preferred starting point.

As far as cosmological SSB goes, many effects are clearly due to virtual energy borrowed from the quantum foam, tipping a precipitation down one particular way and at a particular time rather than down another route. If we go back however to the initial singularity event however, we don't know the explanation yet. Looking for that explanation is one of the Holy Grails of modern physics ....

Fluctuations in the surrounding environment's attributes is one source for the spontaneity.

Observed spontaneity is only illusion because we do not take into account the whole system in all cases.

Spontaneous symmetry breaking happens every time a quantum measurement is performed. A wave function can always (approximatively) be prepared that has the full symmetry of the theory. It is true randomness, not caused by some unknown irregularity. After all, it seems that Nature isn't symmetrical at all.

Hi everyone!

In all the traditional illustrative examples, the same defect as pointed out by Komarov is there. The full details are not taken into account. When everything about the system and its coupling to the entire environment is known, it should be possible to tell which way the symmetry will break. (No quantum indeterminism here!).

Spontaneity is therefore only a kind of an effective theory or explanation, not the final truth.

Can there REALLY be true randomness(Quantum indeterminism) anywhere, as suggested by Masse? This is a fundamental issue with QM. What determines to which state the state will collapse? This is a good point stated by Masse.

Suppose, for an everyday classical example, we say that when a coin is tossed, the outcome is randomly decided between the two alternatives. But the outcome of every toss is amenable to detailed investigation of the dynamics of the coin. Randomness, whether in classical or in quantum situations, is only an approximation and an approach that suits our convenience.

If spontaneity, randomness etc.are accepted as scientific explanation, then obviously, we are back to the good old days of belief-without-investigation! Of course, for a many-body system, or any many-parameter problem, it is difficult to do exact calculations and we need supercomputers of great calculating prowesses to keep track of the evolution of the whole system. And, the randomness approximation comes in handy here.

Can we say the same in case of Collapse? Einstein could not digest this idea of quantum indeterminism due to a fundamental philosophical outlook that searched for order even in the midst of chaos and confusing disorder. Only time will tell which is the correct position, so far QM seems to contradict Einstein though. But with de Broglie-Bohm.theory staging a grand come-back, we can never say that indeterminism (or spontaneity) is a Really Real Reality out there!

The existence of local hidden variables have been dismissed by the Bell theorem and Bell like experiments, which means that it would be true randomness, that is, probabilities don't arise from a lack of information. There are of course other possibilities, but with high costs that entail a drastic change of the theories, and new ways to think about them. It can be for example assumed that the outcome of a quantum measurement depends on the state of the whole Universe. Spontaneity wouldn't then be caused by a local defect, if only symmetries to be broken exist at all.

Claude,

Exactly right - furthermore - hidden variables are in most cases ruled out by Gleason's theorem and moreover, even if for some reason Gleason's theorem did not apply in some exotic cases, it wouldn't help anyway: Renato Renner and Roger Colbeck in Zürich have convincingly demonstrated that no possible hidden variables could possibly improve the significant outcomes produced by current theories

I have a more pragmatic answer to this question. Words like "spontaneity" is nothing other than a folklore! You first need to define the mathematical model intended to describe a given physical system. In QFT or Statistical Mechanics, this definition needs to deal with the thermodynamic limi / ultraviolet limit or both, of a probability measure defining the model. Having the renormalization group formalism in mind, you generate effective potentials associated to the measure. If the limiting measure exists, it can have more than one ergodic component and they may be disjoint. The original symmetry, in the cas it is said to be broken, is not there, as these disjoint components may refer each one of them to a different physical state. Take for instance the Ising model. You have 2 minima. As you grow up the distance scale, in order to reach the thermodynamical limit (infinite volume), there is an effective potential of the double well kind, with a bump separating the minima which grows with increasing distance scale. In the thermodynamic limit, the Landau type potentialevolves to two disjoint parabolic like potentials. If an external or boundary condition is given such that you force the system to be in one of the two minima, it cannot leave there since there is an infinite barrier!

It is still very unfortunate, in my opinion, that some particle physics books still introduce symmetry breakdown by doing a finite shift in the (infinite ranged) field variable in the functional integral. The finite shift is just a redefinition of the integral variable and does no physics.

If you want a nice symmetry breakdown example of day-to-day life, here it is. Take a T-shaped pipe. Add a birthday baloon on each side of the T. Hold one with your hand while you pum air in the tube. Ask the question: if I release the held baloon what will happen? Answer: if the preassure is small, preassure will equilibrate and the empty baloon will fill. But if you pump air over a threshold preassure (small one!), when you release the held baloon it will remain empty! This is the way it is. Symmetry is not present since the two baloons are not identical regarding elasticity. Everything is spontaneous because nature is spontaneous.

The term "spontaneous symmetry breaking" is used very simply to indicated that there is no symmetry breaking whatsoever in the Lagrangian, or Hamiltonian, or equations of motion, which are symmetrical, but the "ground state" is not symmetrical under the full symmetry group. This makes it non-unique, in the classical sense, since we have a whole orbit of equally possible "ground states", having exactly the same energy. This is used in distinction to "explicit symmetry breaking" which is used when an extra term, usually small with respect to the main part of the Hamiltonian, is introduced into an otherwise symmetrical Hamiltoninan, which somehow breaks the symmetry as, for instance in the Zeeman effect, where the magnetic field breaks the rotational symmetry, and hence splits the atomic levels.

Dear Rajat Pradhan,

of course there is spontaneity in your world: Imagine you have the natural law x*x=1, but you live in one world (solution) only, e.g. x=1. Then YOU are just observing a part of the whole picture, and this personal selection from a larger set makes the break. The fact that you were born in the last century is a "spontaneous symmetry breaking". If you want to circumvent such a situation, you have to get reborn all the time, continuously. Of course such fundamental questions are highly philosophical, are also related to many-worlds theories, and I am not an expert of the Bhagavad Gita!

Sincerely yours, Andreas (spontaneously living in Basel, Switzerland).

John's response is the right explanation of what spontanous symmetry breaking stand for. A good example from our daily experience is a round dinner table. At some point you have to make the choice whether you take the glass to your left or to your right. Once someone selects one, all others at the table have to follow. The left-right symmetry is broken. :)

I am afraid that Andreas example misses the point. Take e.g. the case of chiral molecules occurring in living systems. Let for instance x correspond to the operation of inversion (parity), which commutes with the electromagnetic Hamiltonian, H. Then the reduced Cayley-Hamilton Equation equation for the parity operator reads x·x=1, with the eigenvalues being ±1, corresponding to even and odd solutions of the Schrödinger equation (since the commutation relation ensures simultaneous eigenfunctions). However if we add an electroweak perturbation to H, then the solutions exhibits a so-called de lege parity violation. I am using the nomenclature of Martin Quack, see e.g. Handbook of High-resolution Spectroscopy, Edited by Martin Quack and Frédéric Merkt, Wildy (2011). However, one may find localized solutions (classical or quantum mechanically) which do not show the symmetries of the Hamiltonian. They are said to be broken spontaneously. Of course the quantum mechanical superposition principle guarantees that we also have the symmetric state of positive (negative) parity. On the other hand the localized state is broken de facto by a chosen initial condition and time evolved to give an observable probability density. So the three types of symmetry breakings reflect three fundamental hypotheses of the physical-chemical origin of molecular chirality, and it is an open problem to experimentally distinguish between them. For recent developments see the article Fundamental Symmetries and Symmetry Violations from High-resolution Spectroscopy, by M. Quack in the Handbook quoted above.

assuming that there are several solutions of space time developing from a start event

several "end" events can be reached after a fixed time, needing a fixed amount of energy transforming from one inertial system to the other. (e.g. the end event is the same as the start event, except one electron there is present with higher momentum.. mathematically is the same as the change of inertial system for this one particle..). Solutions of end events which are "similar" to the original event can be reached within the same timewindow by many paths.. thus it is a "likely" solution. events which are unlikely.. as for exmple in the extreme case where e.g. everything is the same except the observed particle is located infinite far away or with infinite momentum.. then there is just one possible path to this solution. Now considering the number of eventpaths form one event to superposed end events.. the probability to measure one specific end event will be correlated to the number of possible ways there..

Further thoughts brought me to the conclusion, that all the solutions must have been already present "before" time brings us there.. thus i supose we are living in a cut through a higher dimentional cube where one cut plane can be considered as ala Einstein "space time" - plane where the other solutions which for us apear as QM possibilities are present besides in neighbour planes which are more or less far away..

the directional "driver" of our drifting plane again can be considered as hidden variable.. however, what i wanted to poin out here is, that i supose, it is likely, (due to obviously measured superpositon) that all solutions are already there and time is an illusion generated by the "speed (likely c)" our cutplane drifts through the space time - possibility cube.

Hi Andreas,

Is it really an accident to be born or be living somewhere at some time? Or, are there deeper reasons yet unfathomed by modern science which could satisfactorily explain such spontaneity/accident as SSB. May be, someday we'll find something more interesting than all our examples of SSB, so painstakingly put forward and so forcefully defended, no matter how helplessly one finds something Godellian staring from the other end of the argument. As per our current scientific understanding, how many such accidents have taken place since that proposed mother of all accidents, the Big bang happened to bring us all up to this point? We should let Hoyle and Vikramasinge keep counting! Since you asked about Gita, --- well, it says "nothing is an accident and there is a reason behind everything." A truly causal approach.

Thank you very much.

Dear Rajat, my best wishes and thanks for the stimulating discussion plus the nice words - there is a lot we do not really understand. If you like some more input, maybe Elitzur's theorem in connection with the Higgs mechanism might be an interesting topic to study. Sincerely, Andreas.

http://en.wikipedia.org/wiki/Elitzur's_theorem

http://prd.aps.org/abstract/PRD/v12/i12/p3978_1

Gauge fields obey the original symmetry. Scalar fields obey a

reduced symmetry. Gauge fields spontaneously interact with scalars because

they are not singlets. This spontaneous interaction between scalars and gauge

fields generate masses of gauge fields. In this way symmetry breaks spontaneously.

Thus, in summary, we do not put mass terms to gauge fields by hand. We generate it spontaneously by the mutual interaction between scalars and gauge fields.

If masses of the gauge fields are generated in this way, the theory becomes

renormalizable. Where as a cut-off theory is not renormalizable.

Dear Biswajoy,

Let me try to understand your point. You seem to say that because we need (1) a renormalizable theory and (2) particle masses, the symmetry breaks "spontaneously". This is the well-known bottom-up explanation.

My question is about a top-down explanation of the 'spontaneity' that the reason behind SSB. If we can adduce no other causal reason than the one stated by you, then it is really "spontaneous". What is the reason behind SSB, not what follows SSB. This is why I added in the explanation to the question the "non-anthropic" condition. We can always add any number of reasons (like the two cited by you) to justify SSB post-facto. In fact these reasons are effects of SSB, rather than causes.

One further issue is the non-local nature of the choice of the vacuum at all points of space simultaneously, which would mean signal transfer at infinite speed, no matter how small the universe was at the time of SSB. Otherwise, how could the vacuum suddenly assume the same value everywhere, so that all massless particles suddenly become massive all over the universe at the same time? What determines such a happening ? This is the question.

Thanks and regards,

Rajat

My answer did not address these issues. I used the term

spontaneous theory as opposed to a cut-off theory, where masses

of the gauge bosons are introduced by hand, this renders the

theory non-renormalizable.

In fact this was the original motivation for breaking a theory

spontaneously as opposed to breaking it directly by

introducing mass terms of massive gauge fields.

These issues are discussed in standard text books such

as Halzen and Martin.

For me, "spontaneity" occurs when one is missing ``totality''. Let me use a simple example. The poorest man in your wealthy island is not necessarily the poorest man of the world, but you will never notice it if you are isolated enough.

Something similar happens with, for example, "Poincaré-invariant vacua'' for inertial observers. They are "empty'' and stable under Poincaré transformations, but they turn out to radiate as a black body under accelerations and, therefore, they are not so "empty''!. If you enlarge Poincaré to conformal symmetry by adding acceleration boosts (that is, if you explore beyond), then you will realize that Poincaré-invariant vacua are indeed not so empty; actually, they are coherent states of conformal zero modes, which are "invisible'' (dark) to inertial observers, but they turn "bright'' when you accelerate them.

The Higgs mechanism in electroweak theory, despite its success, I think is also phenomenological. It reminds me the ether time. We should explore beyond the "wealthy islands of particle accelerators'' to reach a more fundamental explanation of mass.

Thank you Manuel for your elaboration.

Spontaneous symmetry breaking can be a general

phenomenon unrelated to High Energy Physics.

When equations describing a physical system has a

certain symmetry but solutions of those equations do not

obey the symmetry then in that system, symmetry is

spontaneously broken.

Dear Biswajoy,

First of all, Happy new year to you and also to all people in this thread.

I think it is not so much the solution not having the same symmetry as the equation, but the choice of such a solution by the system to fall to that is termed SSB. My question is, "What determines this peculiar 'symmetry-breaking choice' out of all possible solutions which very much included many that had the same symmetry as the equation?"

Regards,

Rajat

The term "Spontaneos symmetry breaking' is used in statistical physics for the change of symmetry of the ground state of a system due to the change of its TEMPERATURE, i.e., not due a change of a parameter in the Hamiltonian. As a result of this change of symmetry a spontaneos or, equivalently, second order phase transition to a new phase occurs.

Dear Eva,

Probably, temperature should also be included as a parameter in the Hamiltonian, since it has a role in letting the system transit to a new asymmetric phase. In any case, temperature falling below the critical seems to be the reason behind the SSB.

So, it is then not spontaneous ! Well, out of the infinity of asymmetric ground states, it chooses one to fall to and that is perhaps spontaneous. Which one of the infinity of states available does it select to fall to and why ? This is my question.

If there is no answer to this query as yet, can we hope to at least guess some inner hidden mechanism behind the phenomenon?

Regards,

Rajat

The second order or spontaneous PTs are equilibrium phase transitions, where the ground states are in equillibrium at any temperature except of T=T_c.

The non-equillibrium ones belong to another class of PTs -Non-equillibrium ones. .

Dear Rajat,

when following the discussion above, I fear that some fundamental issues concerning the axiomatic structure of quantum field theory are not widely known. Anyway, don't forget Elitzur's theorem: http://en.wikipedia.org/wiki/Elitzur's_theorem .

A gauge theory cannot be spontaneously broken, you can't break something which is not observable. SSB in the Higgs mechanism is rather a gauge fixing by hand with the result that you obtain a perturbatively renormalizable theory. One may argue that some of the particle masses are due to a coupling to the Higgs field, but the opposite is also true: a kind of Higgs field must be present if you presuppose that some of the particles are massive (http://arxiv.org/pdf/hep-th/9702053.pdf) in order to make the theory consistent (i.e. to have unitarity on a physical sector etc.). This is well-known to specialists, but popular science uses pictures which propagate through tough science without being fully adequate.

Spontaneous symmetry breaking is considered because we

do not want to spoil gauge invariance in SU(2)_L x U(1)_Y

theory. Explicit mass terms break gauge invariance.

In hep-th/9702053 (reference given above) an interesting

converse situation is considered. They consider mass terms

which spoil gauge invariance, but then to restore gauge

invariance they introduce the Higgs field. This is an alternate

point of view of which one should be aware of, I think.

Thank you Biswajoy.

I find Halzen and Martin writing at a place " we then break the symmetry spontaneously" ! Is it not a surprising phrase. Are we breaking it spontaneously or is It broken spontaneously ? Which is true ? Is it in the very make up of the scientist who is trying to understand nature to break symmetries spontaneously or is it in the very nature of objective external universe of phenomena to break the symmetries spontaneously ? Who is breaking and Why?

Regards,

Rajat

Imagine a round table with 10 persons sitting and waiting for dinner to be served. Each person has a large yet empty plate in front of them. The waiter starts by serving little rolles on little plates adjacent to the large plates. After the serving is finished everyone has a roll to his left and one to his right in (36°) rotational and mirror symmetry. The first person who chooses from which side to take the role will break that symmetry. If the first person selects to take the role from the left side, all others will have to do the same. That choice is random, and obviously spontaneous.

Now imagine a ball on top of a rotationall symmetrical hill. It is in an unstable equilibrium state and any even so small disturbance will trigger it to roll down - in any direction. The selection of the direction will be random, maybe linked to the kind of disturbance applied.

A Higgs field has a potential term similar to a hill surrounded by a valey like a Mexican hat. The field will "roll down" (tunnel in QFT) into the valey and stay in a stable minimum - in the vaccum state. Again, the selection of the direction into which it descents is random, each has the same tunnetling likelihood.

Spontaneous symmetry breaking marks a process wherein a state of higher symmetry undergoes transformation into another of lower symmetry. Like any other process, SSB needs to be described with a cause and its effect. For the statistical system, the thermodynamics clearly rules the spontaneity. for example, in isolated system (constant energy and volume) it is entropy (accessibility of states) maximization, that rules the spontaneity. For a coherent state undergoing SSB, QM makes probabilistic description wherein the perturbation from fluctuating vacuum field stands as the cause for the process to happen. The evolved energy in the reduction of the the coherent state is ultimately absorbed in vacuum that is made of infinite oscillators.   The cause effect description being probabilistic, moment-wise success or failure of the SSB occurrence is not there on QM. For SSB in cosmic state change the cause itself is not uniquely described.  The concepts of inflationary state, and formation of Mexican hat like potential are useful for description of SSB path but these are not the cause. 

 It transits like that because the low-symmetry state is more stable? What is the reason? For example: even in spontaneous atomic and nuclear transitions what is the reason?

The term was coined to characterize the "Higgs mechanism" in gauge theory. The ground state of a quantum system might not be invariant under the full symmetry group of the system. There can be a critical point of the potential function that is invariant, but is unstable; e.g.. it may represent a local maximum rather than a minimum. The stable critical points, in the case of the Higgs mechanism, are not unique, but form an orbit under the symmetry group, and no element of this orbit is truly "stable", instead there is motion within this orbit that contributes a nonvanishing energy. This is part of what happens in the Higgs mechanism; the motion in the trough consisting of the orbit gives rise to the mass of the Higgs particle. But the other essential part of the Higgs mechanism is that there is a gauge field mediating the interactions, and this too acquires a mass as a result of combining with the Higgs field in the diagonalization of the Hamiltonian. This is why the Higgs particle, through spontaneous symmetry breaking, is viewed as providing the massive components of the gauge field. The term "spontaneous" is perhaps not   well chosen, but it refers to the fact that the symmetry is broken not by the addition of an explicitly asymmetric term in the Lagrangian, but by the fact that the "stable" ground state is not symmetric under the full symmetry group.

Thank you John for coming on this thread.

Even if the ground state had greater symmetry (instead of lesser), the system would still have transited to that.definitely. So something else is at work.

Spontaneity is specific to a measurement (or observation) of an event at a precise mode of acquisition. The nature of measurement is known, and the interactive measurement is assumed as to have an effect on the system as a consequence. Spontaneous symmetry breaking is a concept that does not require a measurement. It is a thought experiment in which we analyze a hypothetical system that does not have a zero energy potential. So, by definition, energy must be defined as a consequence of landscape, topology or dimension, which additionally may have spin and or charge assignments . The first state to describe is a continuous one. It can be single state, or a periodic loop, or a few other pendulum analogous rest states. The question then becomes, are there conditions within this where the components of the model can differentiate into a non-symmetric transform that is unique from all previous states, and characterize this as being irreversible. If it is reversible, we simply had a super symmetry state in the initial consideration. If it is irreversible, what is the consequential state? Can we write a type of thermodynamic statement and identify what conditions and changes occurred? This leads to a practical analysis of symmetry breaking with real implications for physical emulators and maps. For those who state there is more or less symmetry and that is a factor, I would double check your models and make certain you state exactly what scalar you are working at. For quantum mechanics, complexity is the real thermodynamic qualifier (more or less).

I would like to humbly elaborate on the answer of Pr. Harnad and add a comment on the Higgs mechanism that I think is worth spreading. After which I offer a short digression.

If one talks of the notion of "spontaneous symmetry breaking" (SSB) in the ElectroWeak (EW) unification in Standard Model, here is a standard textbook account: The unique and fully SU(2)xU(1)-gauge invariant energy state of the Higgs field is only metastable (the point at the top of the "mexican hat potential"), while there is not one but a continum of lowest energy groundstates (the circle at the bottom of the mexican hat, a SU(2) orbit) each of which is individually not invariant under the SU(2) gauge symmetry. When the Higgs field transitions from its SU(2)xU(1) gauge invariant metastable energy state to one of its non SU(2)-invariant groundstates, it does so "spontaneously", meaning not necessarily acausaly but in a way that we have no way to predict. Hence the SU(2) gauge symmetry is "spontaneously" broken when the Higgs fields selects one particular groundstate among a continum.

This idea of "spontaneously broken gauge symmetry" is taken to be a cornerstone of the EW unification, and Lawrence Krauss even goes as far as to say that this notion sparked the most revolutionary period in XXth century physics (https://www.scientificamerican.com/article/q-a-lawrence-krauss-on-the-greatest-story-ever-told/#). In my opinion these views are widely exaggerated. Indeed there is now quite enough literature showing that a fully gauge invariant account of the EW unification, dispensing altogether with the notion of SSB, is possible and even preferable. My co-authors and I have given a brief account in section 3 of our review https://arxiv.org/abs/1702.02753, see the discussion 3.3 for references (including, Higgs, Kibble, Faddeev...). I add 'preferable' since such a gauge invariant account is in better accordance with "Elitzur's theorem", a rigorous result of lattice gauge theory stating that a gauge symmetry cannot be spontaneously broken.

I would then say that while the idea of SSB is part of the "context of discovery" of the EW unification and has historical relevance, it may/should not be considered part of the "context of justification".

Beyong the special case of the EW model, when one labels physical events "spontaneous", I think it may mean either of two things:

A) One simply understands "random", meaning that in principle the theory stipulates a causal chain but we are unable to keep track of it in practice; like where the first ice cristal will "spontaneously" form in freezing water.

B) One could refer to quantum events that have only a statistical regularity but no known causal chain reliably producing them; like the "spontaneous" emission of a photon by an excited electron in an atomic orbital, or the "spontaneous" (radioactive) decay of a neutron (or down quark) into proton (or up quark), electron and antineutrino.

The term "spontaneous", properly construed in context, actually conceals genuine scientific explanations. It should not be taken to mean "without rhyme or reason", that is "inexplicable" (As our lesser Newton would have thought of the apple falling). Case A has reason (strict causality), case B has rhymes (statistical regularities).

Thank you very much Jordan for your post. It is very interesting to know that without the notion of SSB EW unification can be understood. 

Your end-note : Both case A and case B offer no reason for the spontaneity that is observed in such cases. Randomness of phenomena and statistical nature of quantum events, both point to something beyond what we know at present.

You're welcome Rajat, it's my pleasure to have the opportunity to share here. 

End-note: As I said, in case A there is a reason; a deterministic chain of causal events leading to the observed phenomenon. But in practice we are simply unable to follow the multitude of microscopic details being part of that chain, as is the case e.g when we are dealing with statistical physics where about 10^23 particles would need to be followed. A case A "spontaneity" is an artifact of our inability to keep track of minute details. It is "epistemological", so to speak. I wouldn't say that it point to something beyond our present knowledge. 

It was Einstein's hope to come up with a theory deeper than QM that would be deterministic. But as far as I know no attempt to find a deterministic causal explanation to quantum regularities have been satisfactory: the so-called "hidden variables theories" are refuted by the violation of Bell's inequalities, Bohm's "pilot wave theory" is  fully equivalent to standard QM but the principle of parsimony would favor QM and the theory never developed into an equivalent of Quantum Field Theory anyway. So at the moment, the "spontaneity" (statistical nature) of quantum events seems to be a feature of Reality. It seems "ontological", if I may say.  Any deeper understanding of Nature would seem to need to incorporate that insight from QM. But this is obviously a deep topic of philosophy of physics, so I would not dare to say that the question is settled. 

I just found this new book by 't Hooft which aims at a fully deterministic foundation of QM, à la Einstein:

 http://www.springer.com/us/book/9783319412849