What is passed from life to life in the process of rebirth?

My understanding of the Theravada perspective:

what is passed from life to life in the process of rebirth?

Depending on what one means by "passed", the stock answer is there is nothing passed from one life to another.

If something passes from one life to another (be it memories, consciousness, karma, etc), than that something should be regarded as one's Self. However, the Buddha's admonished (sometimes, strongly, see MN 38) when someone say his teachings promulgate that some substance can be observed to go through rebirths.

However, the Buddha also said we are heirs of our karma:

“ Bhikkhus, beings are the owners of their kamma, the heirs of their kamma; they have kamma as their origin, kamma as their relative, kamma as their resort; whatever kamma they do, good or bad, they are its heirs.

-- AN 10.216

The difficulty with grasping this subject intellectually is the doctrine of anatta/no-self and buddhist doctrine of karma. Thus, the puzzle:

• if there's no substance that persists, how could we say that something passes from life to life?
• But, if there's no such substance, why the Buddha rejected the idea that physical death is simply the end and talked about "beings appearing at X after death"? And what does it mean to "inherit" karma?

Generally speaking, if we say that karma is what is passed from life to life, then we are turning karma into a substance that persists: saying this is roughly the same as making karma the Self, the Soul. Thus, many avoid saying something of the sorts.

Consider Bhikkhu Bodhi's words about the word "rebirth":

The Buddhist term for rebirth in Pali is "punabbhava" which means "again existence". Buddhism sees rebirth not as the transmigration of a conscious entity but as the repeated occurrence of the process of existence. There is a continuity, a transmission of influence, a causal connection between one life and another. But there is no soul, no permanent entity which transmigrates from one life to another.

So, an answer here is that there's no substance that is passed from life to life, instead there is a causal connection, a process of influence. If the fetters have not being put to an end, after death there's continuity of experiencing dukkha, continuity of experiencing samsara.

Rebirth is happens mometerity through the process of Momentary Dependent Origination (ekaccittakkhanika-paticca-samuppada). So you die and reborn multiple times in a split second. At physical death also something similar happens. The aggregates passes away and arise but in this case aggregates pass away in one body and arise in another.

For a more details explanation with additional references please see this answer

Supreme Buddha is the best example of that of a person developing a knowledge which enabled Him to read past and future lives. For skeptics, and for those of other faiths of course, one may have to give examples of what modern psychics have given, or recorded evidence of ghostly phenomena, or of spirit communications that one get to hear or experience, to explain rebirth.

This is never a question for ones who have gone far in this road less traveled, and the very reason that they went the distance is that they accepted rebirth as an evidentially verifiable fact, never for one moment questioned it. One sure thing is no one will ever come forward to prove to the worldlings – those who are absorbed by worldly pursuits and pleasures – and the skeptics, that this doctrine of rebirth can be proved experimentally.It is never going to happen.

These individuals that I mentioned above are the “Arahant disciples” of the Supreme Buddha – the ones who have found out experientially that they will have no rebirth again. It is because these Noble Ones have overcome the cause for rebirth, and the Patticca Samuppada (Law of Cause and Effect) has been defeated. They have had first-hand knowledge that there is no birth due to Bhava, as there is no formation of Karma to result in a birth (“Bhava paccaya jāti” as noted in Patichcha Samuppada) for Arahants. This is why Arahants are completely free from being born again, are the only ones to know of this truth.

As per the scriptures, it is the flux-of-consciousness, from a past life (Viññaña) that goes onto start of another life. Viññaña never exist independently – in and of itself – but always with Nama-Rupa:

'Were consciousness, Ananda, not to fall into the mother's womb, would mind-and-body (nama-rupa) be constituted there?' 'It would not, Lord'. 'Were consciousness, having fallen into the mother's womb, to turn aside from it, would mind-and-body come to birth in this present state?'. 'It would not, Lord.' (D. II. 62-3)

It is erroneous to say that consciousness (Viññaña a) is able to 'break free' from ‘nama-rupa’. (Here nama is used to refer to all aspects of mind except consciousness itself.) Therefore simultaneous with the arising of rebirth-consciousness there come into being mind and body (Nama-rupa). The six senses (Salayatana) are the inevitable consequences of mind and body. Because of the six senses contact (Phassa) sets in. Contact leads to feeling (Vedana). Dependent on feeling arises craving (Tanha). Craving results in grasping (Upadana). Grasping is the cause of Kamma (Bhava) which in its turn, conditions future birth (Jati). Re-Birth is the inevitable cause of old age and death (Jara-marana).

The only "things" that persist from rebirth to rebirth IMHO are the potential effects of unfruited karmas, which together constitute the continuity of the mind stream. These are non-manifest potentials, not "things" (hence the quotation marks). Thus, there is no "thing," no atta that continues from rebirth to rebirth. Since there is no effect without a cause, and no cause without an effect, every intentional thought, word, and deed that has not yet resulted in an effect at death results in rebirth and consequent experience. This continues until (a) non-intention results in the cessation of new karmas, and (b) all prior karmas have been exhausted, either by fruition or extermination through meditation.

In the Pali suttas, I have not read anything about a "life to life" process of rebirth. In the Pali language, the word for "life" is "jiva". Where as the words often erroneously translated as "past lives" are "pubbe nivāsa", which literally mean "previous homes" & is often more accurately translated as "past abodes", as follows:

Ye hi keci, bhikkhave, samaṇā vā brāhmaṇā vā anekavihitaṃ pubbenivāsaṃ anussaramānā anussaranti sabbete pañcu­pādā­nak­khan­dhe anussaranti etesaṃ vā aññataraṃ

Bhikkhus, those ascetics and brahmins who recollect their manifold past abodes all recollect the five aggregates subject to clinging or a certain one among them.

Any kind of aggregate, whether past, future or present, internal or external, gross or subtle, inferior or superior, far or near, all should be seen as it really is with correct wisdom thus: ‘This is not mine, this I am not, this is not my self.’

SN 22.79 translation Bhikkhu Bodhi

Therefore, the many suttas that mention the Buddha recollecting his "past lives" are actually about the Buddha recollecting his "past homes/abodes", which, in respect to suttas such as the Hāliddikāni Sutta & SN 22.79, may have an entirely different meaning to "past life times".

The above said, the general principles in the Pali suttas refer to the "death" ("marana") of one "body" ("kaya") and, if enlightenment does not occur, the "taking up" of (attaching/clinging to) another "body" ("kaya").

The words "marana" & "kaya" do not necessarily mean "physical death" & the "physical body". "Kaya", which also means "collection" or "group", can refer to a certain set of five aggregates (e.g. five aggregates affected by alcohol, which manifest as a happy person or happy being/satta). When those five aggregates or that state of 'being' ('satta'; refer to SN 23.2) ceases, one "kaya" & "satta" dies ("marana) & a new "kaya" & "satta" arises (e.g. a new collection of five aggregates & a new state of being that have a headache & sickness or addiction due to drinking too much alcohol).

In conclusion, in the same way an addiction to alcohol is developed, what passes from one "body" ("kaya"; "collection") to another "body" ("kaya"; "collection") is craving & its associated underlying tendencies/habits ("anusaya") & effects ("vipaka"). The short answer to the question is: "craving".

Thus the Pali suttas state:

Vaccha, when a being sets this body aside and is not yet reborn in another body, I designate it as craving-sustained, for craving is its sustenance at that time.

Kutuhalasala Sutta

What is passed from life to life in the process of rebirth? Kamma. As taught in AN 5.57:

A disciple of the noble ones considers this: 'I am not the only one who is owner of my actions[Kamma], heir to my actions, born of my actions, related through my actions, and have my actions as my arbitrator; who — whatever I do, for good or for evil, to that will I fall heir. To the extent that there are beings — past and future, passing away and re-arising — all beings are the owner of their actions, heir to their actions, born of their actions, related through their actions, and have their actions as their arbitrator. Whatever they do, for good or for evil, to that will they fall heir."

Ven. Nyanatiloka further explained in his "Kamma & Rebirth":

According to Buddhism, there are three factors necessary for the rebirth of a human being, that is, for the formation of the embryo in the mother's womb. They are: the female ovum, the male sperm, and the karma-energy (kamma-vega), which in the Suttas is metaphorically called "gandhabba," i.e. "ghost," or "soul." This kamma-energy is sent forth by a dying individual at the moment of his death. The father and mother only provide the necessary physical material for the formation of the embryonic body. With regard to the characteristic features, the tendencies and faculties lying latent in the embryo, the Buddha's teaching may be explained in the following way: The dying individual, with his whole being convulsively clinging to life, at the very moment of his death sends forth kammic energies which, like a flash of lightning, hit at a new mother's womb ready for conception. Thus, through the impinging of the kamma-energies on ovum and sperm, there appears just as a precipitate the so-called primary cell."

what is passed from life to life in the process of rebirth?

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