"The most beautiful thing we can experience is the mysterious" ~ Albert Einstein

Monday, July 29, 2013


UFOs. Alien Abduction. Yep, I did just go there. Since I own all 9 seasons of the X-files, I figured I have no choice really! Seriously though, I never really looked much into the subject of Alien Abduction before, other than as a source of entertainment due to one of my favorite TV shows. Despite researching some other topics like ESP, NDEs, etc., which our Western culture may consider off-the-wall, I never bothered looking into Alien Abduction in any depth, because I didn’t really think there was anything substantial to it, unlike the formerly mentioned topics. This was until only recently, when I learned of John Mack on an Internet forum for the Skeptiko podcast and learned there is more to the Alien Abduction phenomenon than I originally knew. This post summarizes his best selling book, Abduction, which I recently enjoyed reading.  I have to admit that I have a hard time swallowing some of this material, but dismissing it outright doesn’t feel right now, either.

We do all seem to have socially pre-programmed knee-jerk reactions to certain "taboo" topics that make us wince and immediately dismiss them as preposterous, the second we hear them mentioned.  Alien Abduction certainly qualifies as one of those topics.  Ironically, in today's society, we are increasingly open to the idea that there is a high probability of intelligent life elsewhere in the Universe. Given that we now know there are billions of stars in our galaxy and billions of galaxies in the Universe, odds are pretty good one of those stars has a habitable planet with intelligent life on it.  There is also the infamous saying that alien life may not only be stranger than we imagined, but even stranger than we can imagine.  In light of that, who are we to say in what form, or in what way, another intelligence may choose to first interact with humanity?  So, while reading this blog post, I encourage you to try to set aside biases or predispositions toward the subject for now and just take the material on face value with an open mind.  I'm going to try and do the same while I write this post!  ;-)

Something I found especially interesting is that the Alien Abduction phenomenon is not quite what the popular media (including the X-files) portrays it to be. The abduction phenomenon has a strong relationship with human consciousness and spirituality, which I think is why Alex Tsakiris (of Skeptiko) has implied "following the evidence" led him to this topic. Further study also confirmed some (not very well formulated) ideas I already had on alien intelligence potentially existing in the Universe. Basically, if consciousness evolves, as is evident from a study of comparative mythology, and alien beings exist that are more advanced than we are, they should also be further along on the evolutionary scale. The most advanced notions from our spiritual traditions across the globe should at least hint, in a consistent fashion, at what this alien intelligence, or alien consciousness, might be like. Indeed, a very consistent theme across all abduction stories is a “shift in consciousness”, or a consciousness expanding experience, that resembles very much the consistent experiences reported by mystics and gurus throughout the ages as they explored higher states of awareness. I discussed some aspects of these experiences (like unity consciousness) to some extent in my Middle Way posts, Part I and Part II

John Mack and the Abduction Phenomenon

Getting back to John Mack, he was a professor of psychiatry at Harvard Medical School and founder of the Department of Psychiatry at the Cambridge Hospital. He is also a Pulitzer Prize winning author for his biography of T.E. Lawrence (the character also portrayed in the classic movie Lawrence of Arabia).  Dr. Mack has devoted his career to exploring the question of how our perceptions shape our relationship with each other and with the world, which perhaps made him an excellent fit for eventually exploring Alien Abduction, because this phenomenon may challenge our notions of perception and reality more than any other.  He was just shy of 40 years of experience in the psychiatry field before coming across the alien abduction phenomenon.

Originally skeptical, Mack grew interested when he was introduced to four experiencers (the more politically correct term for an abductee) by Budd Hopkins, who had already studied over 200 cases himself. The sincerity of these four individuals and seeming authenticity of their cases got Mack interested to start pursuing the field on his own.  After counseling over 100 individual cases, Mack has 76 which fit his criteria for alien abduction, without containing any other mental condition that could account for the story.  Mack points out the many ways the alien abduction phenomenon is still a mystery. However, one conclusion that he and others who have been involved in the sessions agree upon, is that something definitely happened to these folks that defies any conventional explanation.  It is not an “act”, or fraud, or the result of any psychopathology, or conventional physical/emotional trauma from something like sexual abuse, or rape.  Although some cases can (and have been) explained away in these terms, there is currently no explanation that can fully account for all the cases, many of which appear to be genuine.

Here is Mack, in his own words, describing the impression abductees give.

"For me and others who have attended the sessions, as well as for abductees themselves, it is this intensity of this recovered emotion that lends inescapable authenticity to the phenomenon.  Something, everyone who goes through these sessions agrees, has happened to these people, whether of not it is possible to identify the source of what has occurred." (1)

"The intensity of the energies and emotions involved, as abductees relive their experiences, is unlike anything I have encountered in other clinical work."   (1)

Before we go further, I would also like to clear up one misconception about aliens:  if they exist, they must be evil.  After watching popular movies like Fire in the Sky (which did not reflect the Travis Walton case very accurately at all) or Independence Day, it's not hard to get this impression.  However, this is not what the actual abduction phenomenon tells us at all.  Even though most abductees go through various levels of physical/emotional stress and trauma, the consistent theme that comes in after working through the experience is a recognition of being part of a process that is important for both species.  In fact, feelings of mutual love/concern and "brotherhood" is often reported.

Scope of the Abduction Phenomenon

The abduction phenomenon may be more widespread than realized. Over the past 25 years, thousands of individuals have reported abduction experiences to UFO investigators and mental-health professionals. However, a Roper poll of “unusual events", suggests that as many as hundreds of thousands of individuals in the US alone may have had an abduction-like experience.  It is a global phenomenon that is very consistent across cultures and age groups.

As part of his research into the Alien Abduction phenomenon, John Mack visited various cultures to obtain there view on the matter.  One person he spoke with was the Dalai Lama.  In Tibet, since they believe the Universe is full of intelligences other then ourselves, some of which interact with humanity, they showed little surprise over "alien abduction".  Mack also found that Native Americans were already familiar with the alien abductions and already called these other beings Star People.

I'm not trying to say let's all start watching the Ancient Alien TV Series on the (pseudo-)History Channel. which from the one, or two, episodes I have seen, was kind of "out there".  However, it is interesting to note that some cultures do report experiences going pretty far back that very closely resemble the modern-day research on Alien Abduction that has been done by guys like Mack.

It may be that alien intelligence has been interacting with humanity in some fashion for a long time.  This seems to be the message coming from abductees, anyhow.  I'm reminded of the conversation between Deepthroat and Fox Mulder from the X-files TV show.  "They're here, aren't they?", asks Mulder.  "Mr. Mulder, they have always been here", replies Deepthroat.  Hehehe!

Common Themes and Aspects of the Abduction Phenomenon

To fully account for the nature of the abduction experience, John Mack gives the following five salient points that would need to be addressed by any theory purporting to explain the phenomenon. (Please see footnotes for source)
  1. The high degree of consistency of detailed abduction accounts, reported with emotion appropriate to the actual experiences reported by apparently reliable observers. (1)
  2. The absence of psychiatric illness or other apparent psychological or emotional factors that could account for what is being reported. (1)
  3. The physical changes and lesions effecting the bodies of the experiencers, which follow no evident psychodynamic pattern. (1)
  4. The association of UFOs witnessed independently by others while abductions are taking place (which the abductee may not see) [I would also add accounting for the fact that abductees are reported physically missing during time of said abduction] (1)
  5. The reports of abductions by children as young as two or three yeas of age .(1)
I would like to go over some of the common themes that span these five points, but I am going to group them under the three classes of information, which were formulated by Mack, that seem to make up the abduction experience.  We'll see that the full range of the abduction phenomenon challenges our everyday notions of reality, or perhaps points out the limited nature of what humanity thinks they objectively perceive and agree upon as our consensus reality.

Category I - This is called the nuts-and-bolts level, or the physical side of things.  It is here that the phenomenon is the most (but not perfectly) capable of corresponding to our consensus reality, or what we currently understand of physical reality.  This covers things like a visual sighting or radar spotting of a UFO;  burned patches of Earth on the ground where a UFO is purported to land; lesions and scars found on the body; and, similar kinds of things - the kinds of things we can measure, detect and observe and attempt to make some sort of sense out of, utilizing our existing scientific knowledge.

Category II - Here the phenomenon looks like it could be understood within our space/time universe if we had the necessary scientific knowledge to do so, perhaps in thousands of years even.  The alien technology required to span interstellar, if not intergalactic space, to arrive at Earth would be an example.  There are highly theoretical ideas out there like the Alcubierre drive (essentially, a warp drive) and wormholes that prominent physicists like Michio Kaku and Kip Thorne (amongst others) think we may be able to harness within a million years!  Other examples of phenomenon reported that fit under this category is switching off the consciousness of folks in the vicinity, suppressing the memory of abductees and some type of mind control, as well as the creation of alien/human hybrid fetuses.

Category III - Lastly is the range of the phenomenon that doesn't seem to fit within our consensus reality, or spacetime.  It is this area of the experience that most resembles spiritual/mystical experiences reported throughout the ages, as well as the more modern Near Death Experiences (NDE) and Psi experiences.  Some of these include: reports of a sense that part of the experience is happening outside space/time; reports of an altered consciousness of vast other layers of reality hidden behind the 'veil'; reports of opening up to, or becoming aware of, the source of Being, or the Divine ground; reports of an identification with the alien intelligence, which seems to create a dual human/alien identity within the abductee; and even reports of a powerful reliving of past life memories.

There are a few aspects of the phenomenon that Mack groups under Category II, but I think can also go under Category III.  These are the creation and staging of physical landscapes that seem to be only partially real and the fact that the aliens are seen to "shape-shift" in an almost shamanistic fashion.  For example, one abductee reported an experience where all she could remember was an almost surreal and vicious dog barking outside her window.  Later under hypnosis, the dog morphs into an alien upon recall and then she is able to recount the rest of the abduction experience.  For this woman, her experience was mostly a traumatic one that was still highly suppressed at the time.  Another man who could remember some adult experiences via conscious recall had a childhood experience that he couldn't completely remember, but felt may have been an abduction experience.  He was near a road when three black motorcycle riders approached him, as he felt himself enter an altered, dream-like state of consciousness.  When the riders reached him he blacked out and came to what seemed like a few moments later, at which point he saw the riders race away at an incredible speed up another road.  Visiting the actual site later, he confirmed that the road they raced away on did not exist in reality.  Through regression analysis the man was later able to recall memories that confirmed there was indeed an abduction experience during the lost time when he "blacked out" and the black riders were actually alien beings.  Another experience that Mack puts under Category II, but I think could also go under Category III, is that many abductees are "floated" (the most common descriptive used) right through walls when they are taken.  Likewise, the aliens are some times seen coming and going right through walls.

Are these shape-shifting experiences real?  How much is dream?  How much is real?  What about these landscapes that are partially real, and partially unreal?  Is our objective reality as solid as we think it is? Or, are we in for an ontological shock from a potential future discovery that advanced species can bend reality at their will?

There is a philosophy out there called Idealism, which purports that physical reality exists in mind, as opposed to our common sense notions that mind exists in physical reality.  Under Idealism, physical reality is sourced by the mind.  Under Materialism, mind is sourced by the physical brain.  Our consensus reality (i.e. the Universe), under Idealism, is a "storyline" in some greater mind and is constantly shaped by that mind, bringing about the time-evolution of the "story".  A theory for UFOs and alien intelligences under the guise of Idealism says these intelligences are so advanced (in technology and mind!) that they are able to alter the storyline of our consensus reality, or even visit alternate storylines (i.e. parallel Universes) via travel, not through spacetime, but through mind!  Pretty crazy, huh?  Here is a video by Bernardo Kastrup that explains this idea more.  As wild and speculative as it may be, it does seem to fit some of the observations gathered from the reports of experiences from abductees.  It also illustrates the old adage that alien intelligence may be even stranger than we can imagine!

Here are some more common themes that jumped out at me, while reading Mack's book.
  • Abductions typically start occurring in early childhood and continue to recur throughout the abductees life. The experience often “runs in the family”
  • Most abduction experiences start with a “shift in consciousness” and end up resulting in a gap of “missing time”
  • A whitish/blueish light appears upon start of abduction, with the abductee ultimately being “floated”  right through walls as they’re being taken, sometimes accompanied by alien beings that also enter/exit through walls.
  • Physical marks are reported, such as nosebleeds, lesions, scars, scoops, etc., which often heal very rapidly
  • Independent UFO sightings have occurred at the time of abduction and external observers have reported abductees missing during the time abduction was reported to have happened
  • A complete and total lack of control, physically and mentally, is expressed by almost all abductees
  • Physical descriptions of the aliens are fairly consistent, especially reports of the eyes and the effects of looking into the eye
  • Intrusive medical procedures are reported that appear to support some type of human/alien hybridization program (yes, this is hard to swallow, and a rather perplexing aspect of the abduction experience).
  • Telepathic contact and information transfer
  • A growing awareness of ecological concerns for the Earth and a strong desire, or new calling, to help do something about it.
  • An expansion, or growth, in consciousness, which ultimately raises an awareness of the interconnected nature of all life and an experience of unity consciousness
  • An awareness of being a willing, or even voluntary, part of something important for both species
  • A human/alien dual identity (another hard one to swallow, but I have a theory on this I will describe below)
  • Personal growth and transformation as a result of the overall set of experiences
  • A recognition that the aliens are closer to the primal source of consciousness, or God.
  • An awareness that our activities on Earth effect the entire Universe

Let’s go into a few of these themes in a bit more detail.

It seems most abductees start having the experience early in childhood and have recurring abductions throughout their lives.  The nature of the experience changes with time.  In childhood the abduction experiences often have a playful feel to them and the aliens are sometimes even viewed as “play friends”. However, the abductions start to take a more serious nature, once late adolescence, or early adulthood is reached.  This is where intrusive medical procedures may start to happen.  Information transfers of a serious nature on the state of the Earth and human consciousness may also occur.  A paradoxical relationship often also begins at this stage.  At the beginning of the transition the abductee could feel betrayed by the alien figures they were fond of in their childhood, with them now appearing cold, serious and more business-like.  The experience usually becomes more traumatic but, upon working through the fear and trauma, a greater awareness is obtained where they feel enraged by the helplessness, lack of control, and humiliation, while at the same time a mutual love and concern is developed and a sense that one is a willing partner in something greater.

The initiation of abduction is often accompanied by a feeling, tingling, or vibrations within the body, which is then followed by a “shift in consciousness”, or an alteration of consciousness. A spouse in bed next to the abductee, or other nearby individuals, may appear to be “shut off”, much to the dismay of the abductee who may be calling out for help. A whitish/bluish light will typically fill the area, usually without creating any shadows. The abductee will then be “floated” out of his/her room and usually right through the wall. Sometimes the aliens will have appeared on scene by this point, also coming right through the walls and exiting in the same fashion with the abductee. Some will remember being taken to a room, or rooms, in a ship, or UFO, but sometimes all that is recalled is a room, or enclosure. A period of missing time is reported when the abductee is finally returned, whether they consciously remember the experience, or not. Usually, the entire experience is not remembered at the time and only recalled through regression hypnosis later. Other times, none of the actual abduction is remembered, and the abductee will only recall an overall feeling accompanied by a “shift in consciousness”, or a spacing-out and then “waking back up” in the same location facing the wrong direction, or in a different physical location altogether, with a gap of missing time that can be anywhere from minutes to hours.

Perhaps one of the most consistent themes is related to the eyes of the aliens and the effects of looking into them. The overall physical descriptions are fairly consistent too, with a few different “species” of aliens being consistently reported. The eyes, however, are almost always reported as large, uniformly deep black and almond shaped. The reported effects of looking into the eyes are fascinating. At first, most abductees are afraid to look into them, because they fear a "loss of  will", or a "loss of their sense of self".  The abductees may even want to be enraged at the aliens by being subjected to the abduction, but looking into the eyes calms them, as if the alien has complete mental control over them, as well as physical control.  This adds to the powerlessness of the experience, where the abductees not only have zero physical control, but even zero control of their own will and feelings.  When the trauma of the experience is worked through, the fear is let go and the loss of control accepted, information transfer of a more positive and informative nature may come, along with a recognition of an interconnected nature and a feeling of profound love coming from the alien being. There is often a flooding of information and visual images, which is very similar to what we would call telepathic contact. I discussed psi in earlier posts and how it has been experimentally verified to be a real effect at the normal levels of expected scientific rigor and beyond.  If aliens are further along on the evolutionary scale, one might suspect they would have very advanced telepathic abilities.  Furthermore, some psi studies have shown that humans may be able to induce a type of mind-to-mind influence over one another.  Once again, in a much more advanced intelligence, perhaps the alien intelligence can do this to an extent reported in the abduction phenomenon.  If one takes the Yoga Sutras of Patanjali seriously, on rare occasions, yogis with remarkable talents have been able to do similar things.

As far as personal growth is concerned, the following gives a summary of John Mack's outline of the process the abductee goes through while working through his/her experience.  (Please see footnotes for source)
  1. "Pushing through" occurs, i.e., fully experiencing the terror and rage associated with the helplessness and intrusiveness of instrumentation on the ships. When this takes place, acknowledgement and acceptance of the reality of the beings becomes possible.  A more reciprocal relationship follows in which personal growth and learning can take place.  From the "ego death" follow other levels of transformation: (1)
  2. The aliens are recognized as intermediaries, or intermediate entities, between the fully embodied state of human beings and the primal source of creation or God.  (in the sense of a cosmic consciousness rather than a personified being.) (1)
  3. The abductees may actually experience themselves as returning to their cosmic source, or Home, an inexpressibly beautiful realm beyond, or not in, spacetime, as we know it. (1)
  4. Past lives are experienced during sessions with strong emotion appropriate to what is being remembered.  (1)
  5. A distinct but important aspect of this kind of transpersonal experience is an abductees sense of possessing a double human-alien identity. (1)
  6. The reliving of abduction experiences leads abductees to open to other realities beyond spacetime.  Realms that are variously described as beyond the "veil:, or some other barrier that has kept them in a "box", or in a consciousness limited to the physical world.  (1)
Upon reading this list, the spiritual, or consciousness expanding aspect, of the abduction phenomenon becomes obvious.  In regards to the double human/alien identity, which is a difficult one to swallow (as if the rest isn't!), I think there is a reasonable explanation.  In yogic lore and other traditions, unity consciousness is mentioned, where subject and object merge and a state of Oneness is experienced.  I think an encounter with an advanced consciousness, such as with these alien beings, could potentially force this experience upon a person, before they are ready to fully interpret, or understand, the experience, thereby creating a mistaken dual human/alien identity.  As I mentioned above, abductees are often afraid to look into the eyes for fear of "a loss of self". Or, imagine suddenly becoming fully psychic and hearing others thoughts and experiencing others emotions, as clearly and distinctly as your own.  Could this become disorienting, causing one to lose track of who is originating which thoughts and feelings?  Could it serve to bring about an identity crisis of sorts?  Thinking along these lines could perhaps explain why aliens "haven't landed on the White House lawn", as it is popularly phrased.  It could very well be dangerous for humanity as a whole to interact with such an advanced consciousness at this point in time.  It could be they currently prefer this lower level of interaction, which sends a message (through the mouths of abductees) and encourages a development of consciousness (which is already underway!) within humanity, helping to lead us down a path to where we will be ready to meet them "face to face".  All speculation, of course!

Also, I would just like to emphasize again how familiar all this sounds when one considers the mystical and spiritual traditions across the globe, which recount experiences of higher states of awareness, as I partially summarize in Middle Way Part I and Part II.  Also, bullets 3, 4 and 6 are very familiar to the experiences recounted by folks who have had an NDE.  I talk more about those here and here.

Ramifications of the Abduction Phenomenon

The abduction phenomenon is a clear challenge to the dualistic thinking in Western Cultures.  Over the course of the past centuries we have segregated reality into two opposite, or dual, realms, between what we consider to be real and what we consider to be unreal.  In general, the physical is regarded as real, while the spiritual is regarded as unreal.  Many folks are religious/spiritual in our society, but we don't dare talk about "taboo" topics like this in public.  Science dictates what is real by what can be measured.  The nonphysical cannot be defined, let alone measured in an experimental setup.  However, the alien abduction phenomenon consists of many subtleties, which cannot be fully resolved into these two different camps - real/unreal, physical/nonphysical, etc.  It fully encompasses and transcends these dualistic notions and it challenges our perceptions of consensus reality.  The full spectrum of the experience spans the subjective-objective gap, where the lines between real and unreal, between physical experience and psychological experience, become blurred.  Indeed, I think it makes it obvious that the bipolar views of physical vs nonphysical is an inherently limited concept that reality is probably not limited to.  The abduction phenomenon challenges us to gain a deeper, more profound view of reality - a view that seems to be very much in accord with the age-old wisdom traditions contained within all cultures.

Of course, there are cultural biases due to our dualistic thinking here in the West.  If something does not fit within our current worldview, or cannot be explained by our materialist sciences, then it must not be real and is unworthy of consideration.  Or, we try to frame the phenomenon within our existing set of limited notions.  SETI may be perhaps the best example.  The Search for Extraterrestrial Intelligence (SETI) assumes that an advanced intelligence may try to communicate with us via the same modes of communication currently used amongst us - electromagnetic signals, such as radio waves.  Given that psi (which includes telepathic communication) does not seem to be limited by distance, or time, and considering the idea that an advanced consciousness may have mastered these skills to levels we can barely imagine, makes contact via electromagnetic signals seem primitive in comparison.  At the very least, it gives these beings more than one option for methods of contact, but we are ignoring all of them except the one that is most obvious to us.

Common (and Unsatisfactory) Conventional Explanations

There are some common conventional explanations for the abduction phenomenon, such as sleep paralysis, masochism, fantasy prone personalities, mass psychosis, hysteria, or just plain hallucination. I'll quote directly from a John Mack article, where he discusses why these explanations don't fit.  (Please see footnotes for source)

"Abduction experiences can be differentiated from traumatic reactions to simple sleep paralysis for several reasons. First, a portion of abduction experiences occur in the waking state, during the day, and these daytime reports show clear similarities to reports that occur at night. Second, experiencers report similar details and similar procedures that go unreported in the media, and these details are present in the reports of individuals across cultures and among many young children. Sleep paralysis cannot account for these common details. Third, experiencers show similar phobic reactions to events and material symbolically linked to abduction material that are not linked to sleep alone. Fourth, in experiencers, insomnia, anxiety symptoms, and nightmares tend to resolve with the conscious processing of abduction material, which would seem unlikely if the traumatic experience were not directly linked to the material. Although it is possible that abduction reports are the result of some altered-state experience, the common symbolic structure of the reports and their links to symptoms of anxiety require an explanation other than sleep paralysis" (2)

"Abduction experiences do not fit masochistic fantasy for several reasons. First, abduction experiencers do not find the experience pleasurable and do not typically seek to repeat the experience. Second, unlike the controlled exposure of a masochistic fantasy, the abduction experience involves an extreme exposure to helplessness and lack of control. Third, contrary to masochism, sexual arousal is hindered, not enhanced, by an abduction experience. The feelings of humiliation and helplessness found in abduction experiences do not serve the psychological needs of our interviewees in ways that would be expected of a masochistic personality or of a masochist" (2)

Similar problems are encountered with the other conventional explanations in ways you can probably already start to have a feel for.

Final Thoughts

Well, that wraps up this blog post.  I still don't know what to think of the whole alien abduction phenomenon, but it is interesting how it ties back to other experiences, such as NDEs and psi, as well as experiences reported out of spiritual traditions across the globe.  It's also fun to explore, at the very least.

Our age-old traditions across the globe tell us that all life is interconnected, So, I think the most telling message from the alien intelligence is that we need to be better stewards to the Earth (and to each other!), because what we do here effects not only the Earth, but perhaps all life in the Universe.  Is it any coincidence that in this age when we are treating the planet harsher than in any other, the phenomenon of UFOs and abductions are happening to such a degree as they are?

I don't know. But, thinking of all that while looking at this latest picture of Earth from Saturn (taken by the Cassini spacecraft) boggles the mind.  It's an amazing thought that what happens on what looks like a piece of dust floating in space, is intimately tied to and can effect all other life anywhere within that infinite sea of blackness and stars.


As the footnotes indicated, I heavily used the following book in writing this blog post.

(1) Mack, John, M.D., Abduction, Macmillan Publishing, 1994.

Also, for the section discussing the common conventional explanations of the abduction experience, I used:

(2) http://www.ufoevidence.org/documents/doc15.htm

Here are some other great John Mack references.


Also, here is an interesting video that covers some of the more famous and intriguing UFO sightings, for your enjoyment!  :)

I recently received Mack's other book called Passport to the Cosmos, so stay tuned for another post covering that one!

Sunday, July 28, 2013

Quantum Field Theory - Refresher - Part I, Klein-Gordon Field

As some of you may know I like to study physics in my spare time, but doing it as a hobby can be challenging.  With an already busy schedule, it's hard to dedicate the time it takes to master the material.  So, it's been an on-again, off-again kind of thing for the past 15+ years.  Unfortunately, it's easy to forget some of the math during the off periods, so a little refresher is needed to get me going again.  This three to four part blog series is me getting refreshed with some basic Quantum Field Theory (QFT) topics, but hopefully written in a way that makes it interesting to others, as well.  As always, I will try to write around the math in such a way that it's not really needed to understand the equations and still get something interesting out of it.  But, QFT gets kind of abstract, so I'm not making any promises!

This blog post is going to introduce what QFT is about, then move on to one of the simplest fields - the Klein-Gordon Field, a real scalar field - and the quantization of that field.  The next posts in the series will get into various other topics, but will more than likely cover things like: the Dirac Equation and the quantization of spinor fields related to the Dirac Equation;  Lie Groups and the geometry of the local gauge invariance found in Quantum Electrodynamics (QED) - the field theory for explaining the Electromagnetic Force;  field Lagrangians based off a more general gauge invariance, or non-Abelian symmetry groups, like the Yang-Mills Lagrangian found in Quantum Chromodynamics (QCD, the strong force).  If some of the physics jargon in this paragraph doesn't make sense, don't worry - it should become more clear through the discussions.

Natural Units

Before we dig into things, let's go over a concept that comes up everywhere in QFT and Relativity - Natural Units.

Despite the plethora of crazy units out there, there are really only 3 basic units: mass (M), length (L), time (T).  All other quantities can be written in terms of these three basic units.  For example, force is expressed in terms of the three basic units, as follows.

There are all sorts of man-made conventions to quantify these three basic units, like meters, inches and miles for length and kilograms and slugs for mass.  However, three fundamental constants of nature allow us to get rid of all man-made system of units.  Let's see how.

First, upon the advent of Special Relativity and discovery of the constancy of the speed of light (c), Einstein realized we no longer need separate units for length and time.  As strange as that sounds, you're already familiar with this notion if you ever talked about light years.  Sounds like a notion of time, but it is really a measure of length.  In fact, because of the constancy of c, if we consistently measure time in terms of length (or vice-versa), c will not only become dimensionless, but will be equal to 1.

(Actually, what's really intriguing is not the constancy of the speed of light, but that there exists a c such that the spacetime interval is invariant under changes of inertial coordinates.  This means that all observers agree upon separations in spacetime, but disagree, in general, upon separations in space and time, separately.  As a result, the speed of light becomes more of a conversion factor between space and time, appearing as a fixed velocity for all observers.)

Newton's discovery of the universal law of gravity brought another constant (G) into the picture.  Comparing the kinetic energy (½mv²) of a particle with mass, m, in a gravitational potential, with the potential energy (GMm/r), we find that the combination GM/c² has dimensions of length.  With two universal constants now (G and c) we can effectively write mass in terms of length (or time).

Planck topped all this off with his discovery of the universal constant of Quantum Mechanics, ℏ.  The Heisenberg Uncertainty principle says ℏ divided by momentum (Mc) is a length.  This means we can equate the two lengths, ℏ/MC and GM/c², and find that the combination ℏc/G has the dimensions of mass squared.  In other words, we can use the three fundamental constants - G, c and ℏ- to define a mass called the Planck mass.

Likewise, we can immediately define a Planck length (again with Heisenberg's help):

and, with a little more help from Einstein, a Plank time:

So, if we consistently measure mass in terms of the Planck mass, time in terms of the Planck time, and length in terms of the Planck length, we can set ℏ, G and c to 1, in an similar fashion to how we set c to 1 above.  This is what is referred to as Natural Units. 

Incidentally, the fact that the Planck mass is so huge is an ongoing mystery in physics and it represents the extreme feebleness of the gravitational force, relative to the other three.  Also, because the Planck mass is large, the length and time are teeny.

The Planck length and time represents the scale at which physicists believe the notions of space and time may no longer apply, with some more fundamental structure to reality waiting to be discovered.  The Planck time is ~5.4*10^-44 seconds and the Planck length is ~1.6*10^-33 centimeters.  Both are very tiny!  Also, when you hear physicists say the known laws of physics don't allow us to go all the way back to the Big Bang (t=0), the Planck length and time is the specific stopping point they are referring to.  The laws of physics, as we know them today, break down past that point.  We need a quantum field theory for gravity - called Quantum Gravity, for short - to go further.  This is still a work in progress.

Getting back to the Natural Units, remember that all quantities can be written in terms of mass, length and time. Nature has provided us with a set of natural units to measure her by, via the three fundamental constants.  Some folks like to say this suggests we have discovered all the fundamental principles of nature, but I think that is going too far.  There are good reasons to suspect there is more to nature.  In addition, there is gathering evidence that the three fundamental constants just mentioned may not really be constant!

Why Quantum Field Theory

Quantum Field Theory handles both the domain of the very small (Quantum Mechanics) and the very fast (Special Relativity), therefore it is somewhat of a "marriage" of those two separate theories.  There are additional points that makes Quantum Field Theory important.

(1) The combination of Quantum Mechanics with Special Relativity implies that particle number is not conserved

We verify every day in particle accelerators that particles can be created and annihilated, but neither QM nor SR have the theoretical, or mathematical, framework for handling this.  To further illustrate the problem consider a particle trapped in box of size L.  The Heisenberg Uncertainty principle says that you cannot know both the position and momentum of a particle simultaneously with infinite precision.  This is expressed mathematically as:  Δp ≥ ℏ/L.  In a relativistic setting, this implies an uncertainty in energy of: ΔE ≥ ℏc/L.  But, when we have an uncertainty in energy equal to twice the mass of the particle (ΔE = 2mc²) we can start to pop particle antiparticle pairs out of the vacuum, because in relativity we have a mass-energy equivalence.  In other words, if you try to pin down a particle's position too far, you can no longer guarantee you are working with just one particle.  The lower limit on how well a particular particle can be localized is called the Compton Wavelength, defined as: λ = ℏ/mc, where m is the mass of the particle in question.  This immediately tells us that any equation dealing with only a fixed number of particles is doomed to failure in the realm of the very fast and very small.  The Schrodinger equation of QM is such an equation.  QFT offers a way to handle regimes where particle number is no longer fixed, as we'll see later on.

(2)  Because all particles of the same type are the same (duh, right?)

This is more profound than it sounds.  There are two types of elementary particles in the world: bosons (integer spin) and fermions (half-integer spin).  Elementary particles are particles that are not known to have any substructure, unlike say the proton and neutron, which are thought to consist of three quarks each.  It is believed that the quarks are, however, elementary particles and they are classified as fermions, like the electron, the muon, and the neutrinos. Elementary matter particles like the electron are grouped under the fermions.  The force carrying particles like photons (mediator of the electromagnetic force), W and Z bosons (mediator of the weak force) and gluons (mediator of the Strong Force) are grouped under the bosons.  Although a QFT for gravity has not been fully worked out, it is believed the graviton will be the mediator of that force. (The graviton is distinguished from the other fundamental bosons, in that it has Spin 2).  In each case, the particle is the smallest ripple that each "force" field can come in.  Therefore, the graviton is the smallest ripple spacetime - the field in gravity - would be able to come in.  Weird, huh? 

The important point here is that every electron is exactly like every other electron in every possible way!  Within each class, or type, of particle, every particle is utterly indistinguishable from each other.  Swapping two particles around will leave the existing state completely unchanged - apart from a minus sign, in the case of fermions.  This minus sign makes for different statistics between the two particle types. It reflects the fact that any number of Bosons can occupy the same quantum state, but no two fermions can do the same.  This is what makes chemistry (and the reality we know and love) possible, since chemistry depends on where electrons reside within specific atomic orbits, or shells, which are specific quantum states.  If electrons could share the same quantum state, they could all just cascade down to the ground state, or lowest orbit of the atom, and we wouldn't have any chemical interactions!  This is all related to the Pauli Exclusion principle, which you may have heard of before.

Anyhow, with QM, these statistics must be put in by hand, but they fall out of Quantum Field Theory in a very natural way, in something called the Spin-Statistics theorem.

(3)  Causality

To see what Quantum Field Theory offers in the way of enforcing our common sense notions of causality, let's take a look at the amplitude for a particle to propagate from point one point to another, which is given by:

This equation is using Dirac's Bra-Ket (< | | > ) notation, which is common throughout Quantum Mechanics and is a way to represent abstract vectors (and dual vectors).  In Quantum Mechanics, the state of a particle is represented by just such an abstract vector, called the state vector, which "encodes" all the statistical information for all possible observables (like spin, or position).  This state vector lives in a special mathematical space called a Hilbert space, which is an infinite dimensional, square-integrable, complex vector space.  Square-integrable just means the state-vector can be normalized in such a way that the probability interpretation of QM makes sense.  The time evolution of the state of the particle is represented by rotations of this state vector within Hilbert Space.  For each observable, the state vector can be expanded within what's called the eigenbasis for the operator representing that observable.  You can think of these as axes, or dimensions, each of which represents a possible outcome upon measurement.  The projection of the state vector along a specific axis, or eigenvector, quantifies the probability of obtaining that specific outcome upon measurement.  To be specific, the probability is the modulus squared of the projection of the state vector along a specific eigenvector.  The only possible values the projections can take are the eigenvalues of the specific eigenvectors of that operator.  Despite being a complex space, these eigenvalues are guaranteed to be real (by the fact that quantum operators are Hermitian operators, which means the transpose conjugate of the matrix representing the observable is equal to the original matrix).  Also, it's obvious an observable like position has an infinite number of possible outcomes, which is why Hilbert space needs to be able to accommodate an infinite number of dimensions.  (If you don't need [or want!] a refresher on technical QM jargon, feel free to ignore that paragraph and the following one!)

Continuing on, the expression above is called the propagator and the exponent is the unitary time-evolution operator.  A unitary matrix is one where the transpose conjugate of the matrix equals the inverse of the matrix.  These are the complex-space analog of rotation matrices found in every-day normal (i.e. real) Euclidean space.  In other words, a unitary matrix performs a rotation in complex space, just as orthogonal matrices perform rotations in regular space.  So, the propagator takes some initial state-vector and time evolves it to some final state, by rotating it through the complex Hilbert space, via the unitary time-evolution operator, as outlined above.

Getting back to the expression above, we note that in a non-relativistic setting, E=p²/m.  Inserting a complete set of momentum states and performing the integral, ultimately gives:

The only thing important to note here for our purposes is that this expression is non-zero for all x and t, which means that a particle can propagate between any two points in an arbitrarily short amount of time.  It doesn't get better when using the relativistic expression for E, or energy. This is a clear violation of causality in a relativistic theory, because it would allow for faster-than-light signal propagation.  This is bad news in relativity because, if propagation speeds greater than the speed of light are allowed, a reference frame can always be found where the effect proceeds the cause (e.g. you arrive back home, before ever leaving!).  To preserve sanity in the Universe, we like to avoid situations like this.

QFT solves this causality problem one finds in QM (under a relativistic setting) in a rather amazing way - it postulates the existence of anti-particles!  Even in QFT, the amplitude for a particle to leak outside the light cone (i.e. faster than light propagation) is small, but non-zero.  But, this amplitude is canceled exactly by the amplitude of the particle's corresponding anti-particle.  QFT solves the causality problem by introducing anti-particles, or rather nature contains anti-particles to preserve sanity in the Universe!

Klein-Gordon Field

Before we dig into the Klein-Gordon equation and corresponding fields, we need to quickly derive the Euler-Lagrange equation for a field.  This equation is useful for obtaining the equations of motion for a system from its Lagrangian.  Although Lagrangians come in all sorts of complicated scary looking forms, they basically boil down to kinetic energy minus potential energy.  The Lagrangian is a kind of "global" summary of the dynamics of the system and is a very useful expression in Quantum Field Theory for several reasons.

Symmetries and their associated conversation laws play a central role in QFT and these are handled most efficiently within the Lagrangian formulation.   As we'll see below, the Klein-Gordon field we'll be investigating is a free field, which means that particles don't interact with each other.  It's by adding non-linear terms to the Lagrangian that give rise to particle interactions.  There is also an area of study in QFT called the renormalization group, where one can investigate the parameter space of all possible Lagrangians to show that whatever form the unknown theory at higher energies takes (remember, from above, that QFT is valid only up to some value of momentum/energy), it should reduce to what's called a renormalizable field theory.  Renormalizable means we can handle the nasty infinities we talked about above, which rear their ugly heads everywhere in QFT.  So, basically, the Lagrangian formulation is very insightful and useful in QFT and, therefore, is the preferred way to go.  If you don't like looking at Lagrangians, you're going to hate Quantum Field Theory!

A fundamental quantity in classical mechanics is the Action (S), which is the time integral of the Lagrangian (L).  The Lagrangian approach to classical mechanics is a global approach in that it tries to determine the entire trajectory of a particle in one stroke, rather than concern itself with infinitesimal time variations like in the Newtonian Approach.  The Action for a field is given by the following formula. Since we are working with fields, notice that our dynamical variable in these equations is the field (φ).  In Quantum Field Theory, position, which is normally a dynamical variable in classical mechanics, gets demoted to a mere label, similar to time.  The field itself becomes the prime dynamical variable in QFT.  Note also that we are working with the 4 dimensions of spacetime, as well. 

The cursive L is the Lagrangian density, or the spatial density part of the total Lagrangian.  Note that S is a functional.  Whereas a function takes a variable as its argument and returns a number, a functional takes a function as its argument and returns a number.  Hamilton's Principle of Least Action says that when a system evolves from one configuration to another, it does so along the path of least action, or an extremum of S.  This is one of those mysterious facets of nature.  Why does nature behave this way?  Nobody truly knows.

If we vary S and set the first-order contributions to zero, this will give us an extremum of S and provide us with an expression for the equation of motion for the system in question.  The first-order changes in the Action are given by the next equation. 

We need to pull a δφ out of the second term in brackets, so that we can factor it outside the brackets along with the one on the first term.  This is accomplished by integrating the second term by parts, leaving us with:

In order for this to vanish for any arbitrary change in the field (φ), the expression in the brackets must vanish, leaving us with the Euler-Lagrange equation, in this case for a field.

Now, let's consider the following Lagrangian for a real scalar field, or our Klein-Gordon field.

Where we are using the Minkowski metric of Special Relativity, defined by:

It's not too uncommon to see the minus signs swapped around, but as long as one is consistent it all works out.

Let's now plug this into the Euler-Lagrange equation.  Taking the derivative with respect to the field (φ) is straightforward giving us a (-m²φ) term.  The second term is a bit trickier because it requires managing the indices a little bit.  It goes like the following.

Plugging into the Euler Lagrange equation, we get the following:

This is the Klein-Gordon equation, which is the equation of motion for a real scalar field.  Again, note that the prime dynamical variable in this equation of motion is the field.

Quantizing the Klein-Gordon Field

Everything we have done so far is still for a classical field.  Ultimately, to handle the creation and annihilation of particles and the interactions between the particles of the field, we want a quantum field. After all, a particle is the smallest "ripple", or the smallest "quantum", that a particular field comes in!

Before we take a look at how to quantize the Klein-Gordon field, it will first help to review a simpler problem from Quantum Mechanics - the harmonic oscillator.  After seeing how to quantize the harmonic oscillator, we'll proceed unto the more complicated, but analogous, procedure for quantizing the Klein-Gordon field.

You can think of the harmonic oscillator as the quantum version of a classical spring, which is the type we're all familiar with.  As you picture some spring bobbing up and down, you know it can have any old energy, or it's energy spectrum is continuous.  However, for a quantum string the energy states are discrete, with energies in between being forbidden.  In other words, you'll never see a quantum spring take on those energies.  (Of course, on a fundamental level even classical springs are dictated by the laws of quantum mechanics, but on a macroscopic scale the discrete nature of quantum mechanics appears continuous.  It's only by "zooming" way in ... wayyyyy, way in ... that they become obvious)

We need to start off with the Hamiltonian for the Quantum Harmonic Oscillator.

where p and q are momentum and (generalized) position, respectively, and omega is the frequency.  The Hamiltonian is basically the total energy of the system, or kinetic energy plus potential energy.  What we need to do is find the spectrum of this Hamiltonian, or the allowed energy values that the quantum system can take.

What's usually done now is to introduce what are called ladder operators in QM.  Ladder operators step up, or step down, all the allowed energy values for the system in question. We'll see below that there are analogous operators in QFT called creation and annihilation operators that create and annihilate particles.  This was what we were looking for above when we mentioned that particle number is not conserved.  These operators will give us the flexibility to handle the creation and annihilation of particles.  For a very rough analogy, you can imagine a field as a mattress made of springs stretched out through space.  These springs can take on some minimum energy value, which represents a particle, or the smallest allowed ripple/wave of that field.

Here are the ladder operators for the QM version of the harmonic oscillator:

These equations can be reversed to find q and p, in terms of the ladder operators, as follows.

In QM, one encounters something called the commutator, which goes like [a,b] = ab - ba.  The commutator for the ladder operators is:

The relation is important, because we'll also have similar (but more complicated) expressions for the creation/annihilation operators in QFT.  Putting this commutation relation to use, we can rewrite the Hamiltonian in terms of them.

We can also show that the commutator of the Hamiltonian with each ladder operator gives the following.

Now, by acting on the energy eigenkets (|E>) and using the commutation relations we have established so far, we can see that the ladder operators step up and down in energy in steps of omega. (We also use the Schrodinger Equation in the following abstract form H |E> = E |E>.)

It may not be obvious but we have found the spectrum of allowed energy states for the quantum harmonic oscillator.

These are the discrete energies our "spring" is allowed to have.  Normally, this is bounded from below by a ground state, like in the quantum harmonic oscillator, but we won't cover that here.  I just wanted to introduce the concept of ladder operators, or creation/annihilation operators in this more simple form, before moving on to the more complicated looking QFT operators. 

Let's take a look at those now and begin quantizing the Klein-Gordon field.  First, we note that the general solution to the Klein Gordon equation is the following:

You can check this by plugging it back into the Klein-Gordon equation and make sure things work out.  Also, this solution was fixed to ensure that it is always a real value.  In analogy with the equation for p in the quantum harmonic oscillator, we write our conjugate momentum, as follows:

The creation and annihilation operators for our Klein-Gordon field are analogous to the ones for the quantum harmonic oscillator above and given here:

Now that we have the commutation relations between the creation and annihilation operators, let's work out just one of the commutation relations for the field and its conjugate momentum, so we can see how the math goes. 

The second line writes out the commutation relation.  Note how we used p to label the momentum for the field term and q to label the momentum for the conjugate momenta term.  The labels are arbitrary, but they must be different for things to make sense.  A few steps are skipped, but after multiplying all the terms out they can be grouped up via the commutation relations for the creation and annihilation operators.  The first and last commutator vanish, by our first proposed commutation relation stated above.   The second and third commutator can be reduced to three dimensional Dirac delta functions by the second commutation relation, as follows:

Performing the q-integral will basically turn all the q's in the exponents to p's, per the usual behavior of a dirac delta function under an integral.  This gives the following.

Using the standard integral formula for Dirac delta functions and the fact that a constant can be pulled outside a Dirac delta function by multiplying by the inverse of the absolute value of that constant, gives the final result above.  This basically says the commutator is equal to i, if x is equal to y, and otherwise vanishes.

Now, the formula for the Hamiltonian of a field, which we won't bother deriving, is

As you can see there will be very similar math to the commutation relation we just calculated above, however there is probably a couple pages of math that I don't want to put up in explicit detail.  If you can work through the commutation relation above, the Hamiltonian is doable.  I'm just going to quote the final result here and discuss some of the implications:

On the second line above, we simply used the commutation relations again for the creation and annihilation operators.  When this second term is integrated it's going to blow up, per the usual Dirac delta function behavior.  QFT is rife with infinities and this is typically the first of many that one encounters.  Let's explore what's going on.

There are really two infinities hiding in that term.  The first is due to that fact that we're integrating over all space and calculating total energy, when we should be going after energy density.  This kind of infinity is called an infrared divergence.  If we corrected that expression for energy density, we are left with a term that is a sum over all modes of the zero-point energies of the field.  However, this term still blows up as p (momentum) goes to infinity.  Integrating over all values of p is assuming that our theory is valid to arbitrarily high energies, but we know QFT is valid down to some minimum length scale and correspondingly to some finite momentum, or energy scale.  This kind of infinity is called an ultraviolet divergence and was caused here by our own hubris.

There is a real easy way to handle this infinite term.  Ignore it.  In physics, it's only differences in energy from the ground state of H that can be measured, so this infinite shift in the energy density cannot be detected experimentally (as far as we know!).  Therefore, we can just drop that term.  Easy enough.  But ... the rest of the infinities one encounters in QFT aren't quite so easy to get rid of!

There is one big caveat to dropping that term.  Gravity is supposed to see everything!  The sum of all the zero-point energies of the field should contribute to the stress-energy tensor in Einstein's equation for General Relativity (GR).  In GR, it's not just the mass that effects gravity, but the total local energy content, which is recorded in the stress-energy tensor.  This energy would show up in a term, which you have probably heard of before - the cosmological constant.  For some reason, these energies either don't contribute, or cancel to a high degree of accuracy. This is known as the cosmological constant problem.  If you would like a Nobel Prize, you could always try figuring that one out  ;-)

So, we have successfully (and partially!) quantized a real scalar field (the Klein-Gordon field) and we have found the Hamiltonian and the creation and annihilation operators.  However, you may be wondering what about the antiparticles (and causality!)  For a real scalar field, each particle is it's own antiparticle.  It's only when we start to consider complex fields that we will get distinct antiparticles for each particle of the field and the corresponding distinct set of creation and annihilation operators for creating/annihilating those antiparticles.

Well, I guess this post got a little heavy.  I think before continuing on to the next Quantum Field Theory post, I will probably do one that's a little more fun next time.  Stay tuned!